Plausibility: dimensionless groups, characteristic scales, and magnitude bands

Dimensional analysis proves a calculation is consistent. It cannot prove the answer is possible. A cell 2 m across, a Reynolds number of 4×10⁷ in a capillary, and a diffusion time of 300 years across a lipid bilayer are all dimensionally impeccable, and a unit-checking library will pass every one of them.

The three checks below close that gap. scripts/check_plausibility.py runs all of them and verifies dimensional consistency of each formula before reporting a number.

1. Choose the characteristic length first

The single most common error in this whole area is not an arithmetic slip — it is using the wrong length. The dimensionless groups are only meaningful with the length the correlation was fitted against.

Geometry	Characteristic length
Flow in a circular pipe	inside diameter, not radius
Flow in a non-circular duct	hydraulic diameter `4A/P`
External flow over a plate	distance from the leading edge
Flow past a sphere or cylinder	diameter
Conduction in an irregular body (Biot)	volume / surface area
Packed bed	particle diameter
Open channel	hydraulic radius `A/P` — note: radius, not diameter

Using radius where the correlation wants diameter puts every threshold out by a factor of two, which is exactly the size of error that survives review.

2. Dimensionless groups and what they gate

Each threshold is a modelling decision boundary: past it, an assumption in your analysis stops holding.

Group	Definition	Threshold	What stops being true past it
Reynolds `Re`	`ρvL/μ`	2300 / 4000 (pipe)	laminar solutions; above 4000 you need a turbulence model
Péclet `Pe`	`vL/D`	≈ 1	below 1 diffusion dominates, so stirring will not help
Damköhler `Da_I`	`kL/v`	0.1 / 10	above 10 the reagent is consumed at the inlet, so the reactor is transport-limited
Knudsen `Kn`	`λ/L`	0.01	the no-slip boundary condition, then the continuum assumption itself
Mach `Ma`	`v/c`	0.3	incompressibility, at about 5% density change
Womersley `Wo`	`R√(ωρ/μ)`	1 / 10	the parabolic (Poiseuille) profile; above 10 the core moves as a plug
Capillary `Ca`	`μv/σ`	≈ 10⁻³	an interface whose shape is set by surface tension alone
Weber `We`	`ρv²L/σ`	≈ 12	drop integrity — above it, aerodynamic breakup
Bond `Bo`	`Δρ g L²/σ`	1	surface tension holding a drop against gravity
Stokes `Stk`	`ρ_p d² v / (18 μ L)`	0.1	the tracer assumption behind PIV and aerosol sampling
Biot `Bi`	`hL/k`	0.1	lumped-capacitance (uniform internal temperature)
Fourier `Fo`	`αt/L²`	0.05 / 1	the semi-infinite solution; above 1 the body has equilibrated
Schmidt `Sc`	`μ/(ρD)`	—	≈ 1 for gases, ≈ 10³ for small molecules in water
Deborah `De`	`t_relax/t_obs`	1	whether the material is a liquid or a solid on your timescale

Womersley takes angular frequency. Pass 2πf, not f. A resting human heart at 1.2 Hz gives ω ≈ 7.5 rad/s, and in the aorta Wo ≈ 20 — firmly plug-like, which is why Poiseuille's law is the wrong model for arterial flow and the right one for a capillary.

The Reynolds thresholds are pipe-flow values. Transition over a flat plate is around Re ≈ 5×10⁵; for flow past a sphere the wake becomes unsteady near Re ≈ 100. The tool reports the pipe classification and says so.

3. Characteristic scales

Scale	Formula	Sanity anchor
Diffusion time	`L²/D`	10 µm at 10⁻⁹ m²/s → 0.1 s
Thermal diffusion time	`L²/α`	same form, thermal diffusivity
Thermal energy	`k_B T`	4.14×10⁻²¹ J at 300 K
Molar thermal energy	`RT`	2.49 kJ/mol at 300 K
Stokes settling velocity	`Δρ g d²/(18μ)`	1 µm bead in water → ≈ 0.5 µm/s
Mean free path (gas)	`k_BT/(√2 π d² p)`	air at 1 atm → ≈ 68 nm
Debye length	`√(ε₀ε_r k_B T / (2 N_A e² I))`	100 mM → 0.96 nm
Capillary length	`√(σ/(ρg))`	water → 2.7 mm

The L² in diffusion time is the whole story of cell biology. Ten micrometres takes 0.1 s; one millimetre takes 1000 s; one centimetre takes 10⁵ s ≈ 28 hours. This is why cells are small, why tissue thicker than ~200 µm needs a blood supply, and why a claim that a molecule "diffuses across the tissue in seconds" is worth checking.

Stokes settling is valid only while the particle Reynolds number stays below ≈ 0.1. Compute the settling velocity, then feed it back into the reynolds group with the particle diameter as the length. If Re_p > 0.1, the drag law is wrong and the velocity is an overestimate.

4. Magnitude bands

These are deliberately generous observed ranges. A value outside one is worth a second look, not automatically wrong — the tool reports questionable inside one decade and implausible beyond it.

Band	Range	Source
Bacterial cell diameter	0.2–10 µm	Milo & Phillips, Cell Biology by the Numbers, ch. 1
Eukaryotic cell diameter	5–100 µm	Milo & Phillips, ch. 1
Cell membrane thickness	3–5 nm	Alberts et al., MBoC 7th ed., ch. 10
DNA base-pair rise	0.32–0.36 nm	Bloomfield et al., Nucleic Acids
Ribosome diameter	20–30 nm	Milo & Phillips, ch. 1
Protein molar mass	5–1000 kDa	Milo & Phillips, ch. 1
Human capillary diameter	5–10 µm	Guyton & Hall, 14th ed., ch. 16
Mammalian body temperature	306–315 K	Guyton & Hall, ch. 74
Resting heart rate	0.7–3 Hz	Guyton & Hall, ch. 9
Blood plasma osmolarity	275–300 mol/m³	Guyton & Hall, ch. 25
Small-molecule diffusivity in water	3×10⁻¹⁰–3×10⁻⁹ m²/s	Cussler, Diffusion 3rd ed., app. A
Protein diffusivity in water	10⁻¹¹–1.5×10⁻¹⁰ m²/s	Cussler, app. A
Dynamic viscosity of water	0.5–1.5 mPa·s	IAPWS R12-08
Surface tension of water	0.06–0.08 N/m	IAPWS R1-76
Speed of sound in water	1400–1560 m/s	Del Grosso & Mader, JASA 52:1442 (1972)
Speed of sound in air	320–350 m/s	Cramer, JASA 93:2510 (1993)
Sea-level atmospheric pressure	95–105 kPa	ISO 2533
Earth surface gravity	9.76–9.84 m/s²	WGS 84 normal gravity
Visible wavelength	380–750 nm	CIE S 017:2020
Non-covalent bond energy	1–40 kJ/mol	Israelachvili 3rd ed., ch. 2
Covalent bond energy	150–1000 kJ/mol	Atkins & de Paula 12th ed.
ATP hydrolysis free energy	40–60 kJ/mol	Milo & Phillips, ch. 4

Compare binding energies against RT, not against zero. At 300 K, RT is 2.5 kJ/mol. A reported binding free energy of 1 kJ/mol is not a weak interaction; it is indistinguishable from thermal noise.

5. The three errors this catches

A quantity of the wrong kind. Kinematic viscosity (m²/s) where the formula needs dynamic (Pa·s) is the classic. Both are called "viscosity", both are tabulated for water, and they differ by a factor of ρ ≈ 1000. The dimensionality check refuses it before any number is computed:

error: viscosity must have dimensionality [mass] / ([length] * [time]),
       but m²/s is [length] ** 2 / [time]

A unit prefix slip. Micro for milli is three decades. The magnitude bands catch it whenever the quantity is one the table knows.

An assumption used outside its regime. Applying Poiseuille's law at Wo = 20, the lumped-capacitance model at Bi = 5, or Stokes drag at Re_p = 30 all produce a number. The group tells you the number is meaningless.

6. Caveats

The thresholds are conventions with soft edges, not physical constants. Re = 2400 in a very smooth pipe can stay laminar; Re = 2000 with a disturbed inlet may not.
Every group assumes the geometry its correlation was fitted for. Check §1 before trusting a classification.
The bands describe typical observed values, not physical limits. Extremophiles, engineered materials, and pathological states legitimately sit outside them — which is why the tool warns rather than refuses.
A plausible verdict means nothing contradicted the tables. It is not a correctness proof, and it says nothing about whether the measurement was any good — for that, see references/gum-methodology.md.

Sources

Checked 2026-07-26:

White, Fluid Mechanics, 8th ed. — Reynolds, Mach, pipe-flow transition.
Deen, Analysis of Transport Phenomena, 2nd ed. — Péclet, Schmidt, boundary layers.
Incropera et al., Fundamentals of Heat and Mass Transfer — Biot, Fourier.
Bruus, Theoretical Microfluidics — capillary number, low-Reynolds flow.
Berg, Random Walks in Biology — diffusion times, the L² scaling.
Phillips et al., Physical Biology of the Cell, 2nd ed. — k_BT as the biological energy scale.
Milo & Phillips, Cell Biology by the Numbers — biological magnitude bands; bionumbers.hms.harvard.edu.
Israelachvili, Intermolecular and Surface Forces, 3rd ed. — Debye length, bond energies.
Cussler, Diffusion, 3rd ed. — diffusivity tables.
CODATA internationally recommended values — reached through scipy.constants, never typed as literals.