This page tests one question across eight standard dimensionless groups: at a fixed value of the group, how much do the accepted models disagree? If the spread is larger than the effect anyone is trying to measure, the group does not determine the answer, and a claim that it does is a relabelling.
The answer is the same in every case: the group is necessary and never sufficient. The spread at fixed value is the footprint of a second parameter that the group cannot contain — flow regime, inertia, accommodation, Prandtl number, back-mixing, boundary condition, disturbance level, or decoupling.
| Quantity | Group | Compared at | Spread at that value | Spread over range | The missing parameter |
|---|---|---|---|---|---|
| Pipe friction factor | Reynolds number Re | Re = 10^4 | 4.94x | 31.62x | flow regime (laminar/turbulent) is not in Re alone at the transition |
| Sphere drag coefficient | particle Reynolds number | Re = 1 | 1.19x | 1.94x | inertia (finite Re) — Oseen is a first correction, not a solution |
| Channel conductance | Knudsen number Kn | Kn = 1 | 6.27x | 17.48x | the accommodation coefficient and the wall model |
| Natural-convection Nusselt | Rayleigh number Ra | Ra = 10^9 | 1.13x | 1.13x | Prandtl number and geometry enter the prefactor, not the exponent |
| Reactor conversion | Damkoehler number Da | Da = 1 | 1.26x | 1.29x | back-mixing / residence-time distribution |
| Onset of Rayleigh-Benard convection | Rayleigh number Ra | onset | 2.60x | 2.60x | boundary conditions — the group cannot contain them |
| Laminar-turbulent transition | Reynolds number | transition | 1.96x | 1.96x | disturbance amplitude and inlet condition |
| Diffusion coefficient | Stokes-Einstein (D eta / T = const) | eta = 1 Pa s | 0.09x | 125.89x | the decoupling of viscosity from diffusion near a glass transition |
At Re = 10⁴ the laminar law gives f = 0.0064 and the Blasius law gives 0.0316 — a factor 4.94. Both are correct in their own regime; the group is identical. The absent variable is the disturbance environment that decides whether the flow is laminar or turbulent at all. Any claim of the form "friction follows Re" is true only after the regime is specified, which is exactly what the group was supposed to supply.
At Re = 1, Stokes and Oseen differ by a factor 1.19. Oseen is a first-order correction, not a solution, which is why the disagreement grows with Re. The group does not absorb the expansion.
At Kn = 1 the slip expression and the free-molecular limit differ by 6.27. The accommodation coefficient is a surface property that no Knudsen number contains.
The two standard correlations differ by 1.13 at Ra = 10⁹. The exponent ¼ is shared; the prefactor is not, and it carries Pr. This is the mildest case in the table — and still a factor 1.13.
At Da = 1, a CSTR gives 0.500 and a PFR gives 0.632, a factor 1.26. The same Damköhler number describes two reactors that differ by the residence-time distribution. Da alone cannot be used to predict conversion.
Critical Rayleigh number is 1,707.76 for rigid boundaries and 657.51 for free ones: a factor 2.60. The dimensionless group is the same quantity in both; the threshold is not. This is the cleanest illustration in the set, because there is no measurement error and no fitting — the difference is exact.
The laminar–turbulent transition is quoted as 2040, 2300 or 4000 depending on source: a factor 1.96. All three are "the Reynolds number at transition". The inlet condition and disturbance amplitude are not in the number.
The Stokes–Einstein relation is the group Dη/T = const. Near a glass transition the fractional form departs by a factor 125.89 across the viscosity range. The second parameter is the decoupling between viscosity and diffusion, which the relation assumes away.
A claimed dimensionless law should report the spread of its predicted quantity at fixed value of the group. If that spread exceeds the effect being claimed, the group is a relabelling. The criterion is cheap, uses no new data, and is almost never reported.
Applied to the table above, no row survives as a standalone law. Applied honestly, that is not a criticism of the groups — Reynolds, Damköhler and Rayleigh numbers are indispensable. It is a criticism of using them as explanations rather than as bookkeeping.
| # | Hypothesis | Test |
|---|---|---|
| P1-H1 | The spread at fixed group value is predictable from the number of omitted parameters, and grows roughly linearly with it. | Count omitted parameters per law; regress spread on count. |
| P1-H2 | Papers that report a dimensionless law with a fitted prefactor have systematically larger unreported spread than those with a derived prefactor. | Sample 50 papers; compare derived vs fitted prefactors. |
| P1-H3 | Boundary-condition-sensitive thresholds (Rayleigh–Bénard, transition) are the ones most often misquoted as universal. | Survey textbooks for threshold values and ranges. |
Status: computational model results, 3 November 2026. No experiment, no data, no measurement. No novelty claimed; no literature search performed. Code and raw JSON accompany this page set.