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

A scale model that only shrinks length solves a different balance. Cut L by ten and the viscous term grows by ten.

Bacteria cannot swim. Inertia-based propulsion, the thing we call swimming, does not work for them. We paddle and coast. A bacterium lives where that coasting is gone, something closer to clear honey. Same Navier–Stokes equation. Change the scale and a different term runs the flow. One number records the swap: the Reynolds number (the ratio of inertial forces to viscous forces).

Drag the grid's Reynolds slider from about 10 up to 1000. The tidy arrows start to wander.

Near 10, viscosity wipes a disturbance before it travels. The arrows lie down like comb teeth. At 1000 they jitter on their own. That is the face of the flow just before turbulence.

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.

The factor 1/Re sets how many digits the viscous term has#

Those four lines use one formula and one $\nu \approx 10^{-6}\,\text{m}^2/\text{s}$ . Whale $2.50\times 10^{8}$ , swimmer $1.50\times 10^{6}$ , fry $1.00\times 10^{3}$ , bacterium $3.00\times 10^{-5}$ . Divide the ends: $8.33\times 10^{12}$ . About 13 digits. The comment already holds the ratio. Nothing new was run.

In the dimensionless momentum equation the viscous term is multiplied by $1/\mathrm{Re}$ . Treat the dimensionless gradients as order one and that coefficient is the size of the term. The bacterium gives about $3.3\times 10^{4}$ . Inertia at 1 means viscosity about thirty thousand times larger. Multiplying through by $\mathrm{Re}$ and dropping inertia is just that factor, written the other way. The whale gives $4\times 10^{-9}$ . Viscosity is then under four billionths of inertia. A bulk average can drop it. The wall cannot. The gradient there is not order one.

The fry is $\mathrm{Re} = 1.00\times 10^{3}$ , so the coefficient is $10^{-3}$ . Inertia beats viscosity a thousand to one. Not zero. The cylinder wake loses left-right symmetry near 40, one digit under the fry and six digits over the bacterium. The slider marks at 10 and 1000 sit in that gap.

$\mathrm{St} = fL/U \approx 0.2$ closes the frequency as $0.2\,U/L$ . $U$ and $L$ are enough. The pitch does not need a new measurement.

Where the wake behind a cylinder stops being symmetric#

Particles, speed, and the obstacle are fixed on the component. What changes on screen is whether a Von Kármán vortex street (vortices that periodically shed behind the cylinder) appears.

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 $U$ and $L$ are enough to estimate the pitch. Inside a pipe the threshold sits much higher, and the boundary layer splits into several bands. That stack is written up in the moment a water jet breaks up.

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. Non-dimensionalization pulls these dials apart from each other. 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.

A scale model that misses Re solves a different equation#

Cut the length by ten in a wind tunnel and leave the speed alone. $\mathrm{Re}$ drops by ten. The coefficient on the viscous term grows by ten. The spot where the full-scale wing could ignore viscosity becomes a spot where the model feels viscosity more.

Dynamic similarity therefore keeps the product $UL$ . Change the fluid, or raise speed in inverse proportion to length. Match only one dial and the others stay wrong. A free surface also has to match the Froude number that shows up in a kitchen sink, and $U/\sqrt{gL}$ fights the $UL$ requirement. The experiment starts by deciding which dial to drop.