# 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](https://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](https://physics.nist.gov/cuu/Constants/) —
  reached through `scipy.constants`, never typed as literals.