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 provide screens for that gap, conditional on the model. 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.

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

These are ratios of physical scales. Listed thresholds are context-dependent heuristics, not universal points at which an assumption automatically becomes false.

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 while microvascular rheology, discrete cells and vessel compliance can still invalidate a Newtonian rigid-tube capillary model even at low Wo.

The Reynolds thresholds are pipe-flow values. Transition over a flat plate is around Re ≈ 5×10⁵; sphere wakes and boundary layers have different geometry-specific transitions. The tool reports the pipe classification and says so.

3. Characteristic scales

L² gives a characteristic diffusion scaling, not a complete transport model. Ten micrometres takes 0.1 s; one millimetre takes 1000 s; one centimetre takes 10⁵ s ≈ 28 hours. These values assume D = 10⁻⁹ m²/s. Geometry, dimensionality, binding, membrane partitioning and reaction change actual first-passage/equilibration times. The scale alone cannot infer a biological size limit or tissue viability.

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, creeping-flow drag needs review, including finite-Re corrections, walls and shape.

4. Magnitude bands

These are illustrative screening bands, not validated reference intervals or precise ranges extracted from each cited textbook. Their endpoints are curated choices. Verify the relevant organism, phase, temperature, pressure, composition and definition before interpreting a flag. Historical textbook sources below are retained as background and were not all independently reread in this refresh. A value outside one is worth a second look, not automatically wrong — the tool reports questionable inside one decade and implausible beyond it.

**Compare binding energies against RT, not against zero.** At 300 K, RT is 2.5 kJ/mol. A 1 kJ/mol free-energy difference still changes equilibrium probabilities by a factor of about exp(1/RT) = 1.5; it is not indistinguishable from noise. Binding standard states, sign, concentrations and cooperativity matter.

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 warns that the model assumptions need review; it does not establish the error in the predicted number.

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 2026-10-01:

• 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.

Current primary checks (2026-10-01):

• NASA Reynolds and Mach definitions.
• IAPWS viscosity and surface tension R1-76(2014): water bands here are near-ambient illustrations, not the full validity region.
• CIE visible radiation: visibility has no precise universal wavelength limits; 380–750 nm is the helper's conventional window.
• Cell Biology by the Numbers: quantities and energy scales depend on the organism, conditions and choice of standard state.

protein_molar_mass expects mass/amount (kg/mol). A bare kDa is molecular mass; convert with N_A explicitly. The ATP band expects a positive magnitude; signed hydrolysis delta G is usually negative. Neither physiological bands nor exit statuses are clinical reference limits. The helper's overall verdict concerns only requested bands, not the validity of computed groups/scales. It uses nominal constants and standard gravity, without propagating their uncertainties.