The molecular limit

Where a real fluid stops following the equations, in metres, seconds and metres per second.

The viscosity rescaling of the scales page makes the core radius $\ell_r = \sqrt{\nu\tau}$ and the core speed $U = \sqrt{\nu/\tau}$, each up to a fixed factor that the theorem never states. Set those factors to one. Then the product $U\ell_r=\nu$ for every fluid, and three questions have numerical answers.

Three coincidences that are one

The continuum description fails when the core radius reaches the mean free path, $\ell_r=\ell_{\rm mfp}$. That happens at

$$\tau^* = \frac{\ell_{\rm mfp}^2}{\nu},\qquad U^* = \frac{\nu}{\ell_{\rm mfp}},\qquad \mathrm{Ma}^* = \frac{\nu}{\ell_{\rm mfp}\,c},\qquad \frac{v_T}{U}\Big|_{\ell_{\rm mfp}} = \frac{1}{\nu}\sqrt{\frac{k_BT}{\rho\,\ell_{\rm mfp}}} .$$

Kinetic theory gives $\nu\approx\tfrac12\ell_{\rm mfp}\bar v$ with $\bar v$ the mean molecular speed, and the sound speed is of the same order as $\bar v$. So $U^*\approx\bar v/2$ and the Mach number at the moment the continuum fails is of order one. In English: the core hits the molecular scale, the molecular speed, and the sound barrier at the same instant, because all three are the same statement about what viscosity is. Incompressibility fails with the continuum. The isothermal assumption fails too, since a Mach-one flow heats.

What to look for: the core speed and the thermal velocity at the core scale both rise as the radius falls, but the thermal one rises faster, so the gap between them closes toward the mean free path. The dashed line is the mean free path for air or the molecular scale for water; to its left the chart describes nothing. If the two solid lines crossed to the right of the dashed line, thermal fluctuations would exceed the flow inside the continuum regime and the deterministic core would be meaningless there; they do not, for either fluid, with unit prefactors.

The numbers

Fluid	$\nu$ (m$^2$/s)	molecular scale	$\tau^*$	$U^*$	Ma$^*$	$v_T/U$ there
air, 20°C	$1.5\times10^{-5}$	68 nm (mean free path)	0.31 ns	220 m/s	0.64	1.5%
water, 20°C	$1.0\times10^{-6}$	1 nm (a few molecules)	1 ps	1000 m/s	0.68	6%

Karniadakis's figure of 70 nanometres for air is the mean free path. The table says what that entails: the equations stop describing air a third of a nanosecond before the singular time, when the core is moving at two-thirds the speed of sound and the thermal velocity at the core scale is over one percent of the flow. Water has no mean free path in the gas sense; a nanometre is where the continuum is conventionally taken to end, and the picture is the same.

Scope

The prefactors are unknown. The core speed is $e_0\tau^{-A}$ with $e_0>0$ a constant of the profile construction that the paper does not evaluate, and the relevant core radius is $\sqrt{2X_{\rm in}\tau}$ with $X_{\rm in}$ a chosen similarity radius. A large $e_0$ makes the speed hit the sound barrier earlier in $\tau$, at a larger core radius; a small $X_{\rm in}$ makes the radius hit the mean free path earlier. Both move the continuum failure away from the singular time, never toward it. The exponent $h$ changes the powers by less than one part in a hundred and does not affect the conclusion. The thermal ratio uses the Landau–Lifshitz estimate of velocity variance in a volume $\ell^3$ and is a scaling argument, not a computation of the noise projected onto the pulse modes; the seeds page takes that step heuristically.

Next: seeds and noise.