Bugs Swim in Honey — The World of Flow Changed by Reynolds Number

The Reynolds number revealed by non-dimensionalization, and how scale creates differences in flow.

Bacteria cannot swim. To be precise, inertia-based propulsion, which we call "swimming," does not work for bacteria. Unlike us, who move forward by paddling our tails and relying on inertia, bacteria live in a world where inertia has vanished—a place much like transparent honey. This post explores why the same Navier-Stokes equations create completely different worlds depending on the scale, and how the Reynolds number (the ratio of inertial forces to viscous forces), which summarizes this difference in a single number, becomes the "hidden grammar" of fluid mechanics. At the end, we also look at why non-dimensionalization—the process of stripping the units from an equation to leave only its structure—is the first grammar of fluid mechanics.

Same Equations, Different Worlds#

Blue whales swim in the ocean. The water molecules are, of course, the same water. However, for a 1 μm bacterium, water feels like a completely different substance. This is because even if the same Navier-Stokes equations are used, the dominant terms change as the scale changes.

The incompressible Navier-Stokes equations look like this:

\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u}

Where $\rho$ is the density, $\mathbf{u}$ is the velocity field, $p$ is the pressure, and $\mu$ is the viscosity coefficient. The left side represents inertia (mass × acceleration), and $\mu\nabla^2\mathbf{u}$ on the right side is the viscosity (internal friction) term. These two forces fight at every moment. Who wins?

Stripping the Equations: Non-dimensionalization#

The answer is determined by the scale. To show this, we must non-dimensionalize the equations. We divide all quantities by a representative length $L$ , a representative velocity $U$ , a representative time $L/U$ , and a representative pressure $\rho U^2$ .

\tilde{\mathbf{x}} = \frac{\mathbf{x}}{L}, \quad \tilde{\mathbf{u}} = \frac{\mathbf{u}}{U}, \quad \tilde{t} = \frac{tU}{L}, \quad \tilde{p} = \frac{p}{\rho U^2}

The meaning of each variable is "measuring that physical quantity by its own natural size." Substituting these definitions into the Navier-Stokes equations and rearranging them leaves a clean equation where all units have disappeared.

\frac{\partial \tilde{\mathbf{u}}}{\partial \tilde{t}} + (\tilde{\mathbf{u}}\cdot\tilde{\nabla})\tilde{\mathbf{u}} = -\tilde{\nabla}\tilde{p} + \frac{1}{\mathrm{Re}}\,\tilde{\nabla}^2 \tilde{\mathbf{u}}

All physical units disappear, and only one number $\mathrm{Re}$ remains. This is the Reynolds number.

\mathrm{Re} = \frac{\rho U L}{\mu} = \frac{U L}{\nu} = \frac{\text{Inertial Force}}{\text{Viscous Force}}

$\nu = \mu/\rho$ is the kinematic viscosity. The larger $\rho$ , $U$ , and $L$ are, the more inertia wins, and the larger $\mu$ is, the more viscosity wins. Even in the same water, if $L$ decreases, $\mathrm{Re}$ drops linearly. Eventually, the terms that govern the world change.

Scale Changes the Reynolds Number#

Let's feel it with numbers. Based on the kinematic viscosity of water $\nu \approx 10^{-6}\,\text{m}^2/\text{s}$ , the $\mathrm{Re}$ of representative organisms is as follows:

# Calculate Reynolds number with representative length/velocity
def reynolds(U, L, nu=1e-6):
    return U * L / nu
 
cases = [
    ("Blue Whale",   10.0,    25.0),
    ("Swimmer",      1.0,     1.5),
    ("Fry Fish",     0.1,     0.01),
    ("Bacteria",     30e-6,   1e-6),
]
for name, U, L in cases:
    print(f"{name:12s} Re = {reynolds(U, L):10.2e}")
# Blue Whale   Re =   2.50e+08
# Swimmer      Re =   1.50e+06
# Fry Fish     Re =   1.00e+03
# Bacteria     Re =   3.00e-05

The difference between the largest and smallest values is a whopping 13 orders of magnitude. And that's in the same water. The Reynolds number is not just a simple number; it's an address that labels different "worlds" even within the same fluid.

In regions where $\mathrm{Re} \ll 1$ , like for bacteria, the inertia term of the non-dimensional equation practically disappears. Multiplying both sides by $\mathrm{Re}$ yields the Stokes equation $-\nabla p + \mu \nabla^2 \mathbf{u} = 0$ . There is no time derivative. In other words, the flow around bacteria "cannot remember the past." The moment the force is removed, the velocity also becomes zero.

Building Intuition through Visualization#

Try manipulating the Reynolds number directly in the simulation below.

If you lower the Reynolds number to around 10, the vector field becomes orderly like a comb. This is because viscosity quickly dissipates all disturbances. Conversely, if you raise it to 1000, the vectors begin to shake irregularly. This is the precursor to turbulence.

The following is the flow behind a cylinder. The point of observation is the formation of the Von Kármán vortex street (vortices that periodically shed behind the cylinder).

At $\mathrm{Re} \lesssim 40$ , the flow behind the cylinder remains attached and symmetric. When it exceeds 40, vortices on the top and bottom shed alternately, showing a pattern where "the tail shakes periodically." Flags fluttering, power lines whistling, and long bridges vibrating in the wind are all due to the same Von Kármán vortex street. The frequencies of natural phenomena are tied to the Strouhal number $\mathrm{St} = fL/U \approx 0.2$ , so if you know $U$ and $L$ , you can even estimate the pitch of the sound.

Why are Bacteria "In Honey"?#

At $\mathrm{Re} \ll 1$ , inertia is close to zero. The moment you stop swimming, you stop. There's an even more surprising result.

You cannot move forward with time-reversible motions — Purcell's scallop theorem.

An opening and closing motion, like that of a scallop, results in zero net displacement. Therefore, bacteria must rotate helical flagella or perform asymmetric whip-like motions to move forward. Starting from the same Navier-Stokes equations, the scale of $\mathrm{Re}$ has changed the "list of allowed movements." The grammar of motion design has changed.

Conversely, when $\mathrm{Re}$ exceeds several hundred thousand, the viscosity term of the equation becomes negligibly small compared to the overall flow (with the exception of the boundary layer). Thus, the average flow around an aircraft wing is fairly well explained by the inviscid Euler equations. The equations haven't changed; only the dominant terms have.

What Non-dimensionalization Tells Us#

The Reynolds number is just one example of a dimensionless number. By the same logic, "world dials" such as $\mathrm{Ma} = U/c$ (Mach, inertia/compressibility), $\mathrm{Fr} = U/\sqrt{gL}$ (Froude, inertia/gravity), and $\mathrm{We} = \rho U^2 L/\sigma$ (Weber, inertia/surface tension) appear continuously. The core of non-dimensionalization is making these dials independent. This is why scale models can be used in wind tunnels for experiments. Even if the size and speed are different, if $\mathrm{Re}$ is the same, the non-dimensional equations are the same, and therefore the solutions are the same. This is the law of dynamic similarity.

Key Points to Remember#

Non-dimensionalization is the process of stripping the units from an equation to leave only its essence. In the place where units disappeared, world dials like $\mathrm{Re}$ , $\mathrm{Ma}$ , and $\mathrm{Fr}$ appear.
The Reynolds number is the scoreboard of inertia and viscosity. Even in the same water, if the scale changes, the score changes, and the face of the flow changes.
The reason bacteria live in "honey" is because of scale. The Reynolds number, which is 13 orders of magnitude lower in the same water, is the evidence and a signal that nature requires different motion designs for each scale.