Scaling and dimensionless numbers

Reynolds, Mach, Strouhal, ka, Weber, Capillary — when each one matters.

A continuum equation in three dimensions can be written down in pages of indices, or it can be rescaled to its handful of natural ratios. Physicists prefer the second route: the dimensionless groups that emerge from non-dimensionalising a system identify, with maximum economy, the regimes of behaviour the system actually exhibits. The Math Foundations chapter on dimensional analysis develops the method (Buckingham π); this chapter applies the method live and catalogues the dimensionless numbers physicists actually reach for.

Why dimensionless numbers exist at all

Every governing equation in this bookshelf has the form

A_1 \, (\text{term}_1) \;+\; A_2 \, (\text{term}_2) \;+\; \ldots \;=\; 0,

with $A_i$ dimensional coefficients (density, viscosity, surface tension, speed of sound). Rescaling each variable by its characteristic value collapses the equation to a dimensionless form whose only parameters are ratios of the original coefficients.

▶ The Reynolds number from non-dimensionalising Navier–Stokes

Start from incompressible Navier–Stokes:

\rho\, \frac{\partial \mathbf{u}}{\partial t} \;+\; \rho\, (\mathbf{u}\cdot\nabla)\mathbf{u} \;=\; -\nabla p \;+\; \mu\, \nabla^2 \mathbf{u}.

Rescale: $\mathbf{u} = U \tilde{\mathbf{u}}$ , $\mathbf{r} = L \tilde{\mathbf{r}}$ , $t = (L/U) \tilde{t}$ , $p = \rho U^2 \tilde{p}$ . Substituting and dividing through by $\rho U^2 / L$ :

\frac{\partial \tilde{\mathbf{u}}}{\partial \tilde{t}} \;+\; (\tilde{\mathbf{u}}\cdot\tilde{\nabla})\tilde{\mathbf{u}} \;=\; -\tilde{\nabla}\tilde{p} \;+\; \frac{\mu}{\rho U L}\, \tilde{\nabla}^2 \tilde{\mathbf{u}}.

The only dimensional coefficient that survives is $\mu/(\rho U L) = 1/\mathrm{Re}$ . The Reynolds number is the unique dimensionless parameter of incompressible Newtonian flow.

PDE:

original (dimensional)

ρ Du/Dt = −∇p + μ ∇²u

choose scales

x ↦ L x˜, u ↦ U u˜, t ↦ (L/U) t˜, p ↦ ρU² p˜

dimensionless form
Du˜/Dt˜ = −∇˜p˜ + (1/Re) ∇˜²u˜

dimensionless groups

Re = ρUL/μ (inertia / viscous)

Non-dimensionalising a PDE turns every dimensional coefficient into a *ratio* of physical quantities, exposing the parameter(s) that determine the system's behaviour. For Navier-Stokes, the only such ratio is the Reynolds number; for the heat equation in a flow, it's the Péclet number; for the linear wave equation, there's no free parameter — all linear waves obey the same dimensionless equation. The number of dimensionless groups is given by the Buckingham π theorem; their *values* select the regime.

The procedure applies to any PDE. The Navier-Stokes equation gives Re; the convective heat equation gives the Péclet number; the linear wave equation has no free parameter at all (all linear waves obey the same dimensionless equation).

The dimensionless-number catalogue

Symbol	Name	Formula	Compares	Where it appears
Re	Reynolds	$\rho U L / \mu$	inertia / viscosity	Navier–Stokes; fluid mechanics
Ma	Mach	$U / c$	flow speed / sound speed	Compressible flow; Sound Ch 9
Ma_a	acoustic Mach	$v'/c$	perturbation / sound	Nonlinear acoustics; Sound Ch 10
Sr	Strouhal	$f L / U$	oscillation / advection	Vortex shedding
ka	Helmholtz	$\omega a / c$	source size / wavelength	Sound Ch 6 sources
We	Weber	$\rho U^2 L / \sigma$	inertia / surface tension	Droplet break-up
Ca	Capillary	$\mu U / \sigma$	viscous / surface tension	Coating flows
Bo	Bond	$\rho g L^2 / \sigma$	gravity / surface tension	Meniscus shape
Pr	Prandtl	$\nu / \alpha$	momentum / heat diffusion	Boundary layers
Sc	Schmidt	$\nu / D$	momentum / mass diffusion	Mass transport
Le	Lewis	$\alpha / D$	heat / mass diffusion	Combustion
Kn	Knudsen	$\ell / L$	mean free path / size	Kinetic theory
Pe	Péclet	$U L / D$	advection / diffusion	Transport in flows
Ek	Eckert	$U^2 / (c_p \Delta T)$	kinetic / thermal	Aerodynamic heating

The Re × Ma plane

The Reynolds number and Mach number together identify, to first order, the regime of single-phase flow:

Re5.00e+4

Ma6.76e-4

We694

Ca0.0139

Speed U = 1.00 m/s Length L = 0.0500 m Fluid:

Scenarios:

The two-axis map locates a flow in the Reynolds × Mach plane. Scenarios spanning fifteen decades of Re all collapse onto this one chart; the regime — Stokes vs. laminar vs. turbulent; incompressible vs. supersonic — is read off by which coloured band the marker sits in. Use the sliders to dial in a custom case, or click a preset to jump to a canonical scenario.

Fifteen decades of Re, eight decades of Ma — most of the bookshelf’s flows sit somewhere on this two-axis map. Click the scenario buttons to jump from a swimming bacterium (Re ~ 10⁻⁵) to a Concorde at Mach 2 (Re ~ 10⁹).

The crossover method — set the ratio to 1

When an equation contains two competing terms, the crossover scale — the length, time, or amplitude at which they are comparable — is given by setting the relevant dimensionless number to 1.

Crossover:

competition

inertial: ρU²L² vs viscous: μUL

setting ratio = 1

Re = ρUL/μ = 1 → L* = μ/(ρU)

crossover
L* ≈ 0.0010 mm

interpretation

Below L*: Stokes flow (viscous-dominated). Above: inertial.

U = 1.000 m/s μ = 1.00e-3 Pa·s ρ = 1000 kg/m³

Every governing equation in this bookshelf has two-or-more competing terms. The dimensionless number that compares them is *the* parameter of the regime: below the crossover (L < L*, v' < v'*, etc.) one term dominates; above it, the other does. The crossover scale is found by simply setting the dimensionless ratio to 1 and solving. This is the most-used estimation technique in physics.

Three examples:

Stokes vs. inertial. $\mathrm{Re} = 1$ at $L \sim \nu/U$ . For air ( $\nu \sim 10^{-5}\,\text{m}^2/\text{s}$ ) at 1 m/s, this is 10 microns — below that scale, flow is creeping.
Capillary vs. gravity. $\mathrm{Bo} = 1$ at $L = \ell_c = \sqrt{\sigma/(\rho g)}$ . For water, 2.7 mm — below that scale, drops are nearly spherical.
Linear vs. nonlinear acoustics. $\mathrm{Ma}_a = 1$ requires $v' \sim c$ , an essentially explosive amplitude. For audible sound the linearisation is overwhelmingly justified.

The crossover scales are not boundaries; they are the natural scales at which the relevant competition between effects becomes interesting.

Three orders of magnitude is the relevant unit

Dimensionless numbers are useful because of their order of magnitude, not their precise value. The crossover from laminar to turbulent pipe flow is somewhere between $\mathrm{Re} = 2000$ and $\mathrm{Re} = 4000$ , depending on disturbance level — but a flow at $\mathrm{Re} = 10$ is laminar and $\mathrm{Re} = 10^5$ is turbulent, and these conclusions do not depend on which side of $2300$ each number sits.

This is also why semilogarithmic ranges (decades, not units) are the natural scale for physical reasoning. The acoustic Mach number for ordinary speech ( $v' \sim 10^{-3}$ m/s, $c = 343$ m/s) is $\sim 3\times 10^{-6}$ — emphatically not close to nonlinearity. The acoustic Mach number near a shock ( $v' \sim c$ ) is $\sim 1$ — emphatically at it. Intermediate cases (a jet engine at $10^{-2}$ , an explosion at $0.3$ ) sit in transitional regimes.

Geometric and energetic scaling

Dimensionless numbers also drive order-of-magnitude estimates that bypass full derivations:

Why a 1 cm raindrop is shaped like a hamburger. $\mathrm{Bo} = (L/\ell_c)^2$ . For a 1 cm drop in air, $\mathrm{Bo} \approx 14$ ; gravity dominates and the drop flattens. For a 1 mm drop, $\mathrm{Bo} \approx 0.14$ ; surface tension dominates and the drop is spherical.
Why the basilar membrane has 100× stiffness variation. A passive resonator at the base must respond at 20 kHz; at the apex, at 20 Hz. The ratio of natural frequencies is 1000, and $\omega \propto \sqrt{k/m}$ , so $k$ varies by $10^6$ in principle. Other factors (mass loading, fluid coupling) bring this down to the measured ~100×.

These scaling estimates give precise predictions about scaling — what changes how fast as you change the system — within a factor of two or three.

⏳ The history — Reynolds, Buckingham, and the rise of similitude

Osborne Reynolds in 1883 published the classic experiment: water flowing in a glass tube, with a thread of dye injected upstream. At low flow rates the dye thread is a straight ribbon; at high flow rates it bursts into chaotic eddies. Reynolds measured the transition and observed that it occurred at the same dimensionless combination $\rho U d/\mu \approx 2000$ regardless of fluid, pipe size, or velocity.

Reynolds’s experimental result was a similitude principle: two flows with the same dimensionless parameter behave identically when expressed in dimensionless variables. This is the practical foundation of model testing — a scale model of an aircraft in a wind tunnel, sized so that $\mathrm{Re}_\text{model} = \mathrm{Re}_\text{full}$ , will exhibit the same boundary-layer separations, the same vortex shedding, the same drag coefficient as the full-scale vehicle.

Edgar Buckingham in 1914 systematised the procedure mathematically: any physical relation expressible in $n$ variables with $k$ independent dimensions can be re-expressed as a relation between $n - k$ dimensionless groups. This is the Buckingham π theorem. Combined with Reynolds’s similitude principle, it gives the modern engineering tool of non-dimensional analysis.

For physics the lesson is more philosophical. The same equation describes E. coli swimming and a 747 wing, with the only difference being which dimensionless number is small and which is large. The dramatic qualitative diversity of fluid flow is not in the equations but in the parameter space they live on.

For the cross-book applications — flow regimes, crossover scales, acoustic Mach number across the books — see the key examples sub-page.