How could we estimate the number of molecules below which a fluid cannot be adequately modeled by the Navier-Stokes equations?

Question

As is known, the Navier-Stokes equations are an approximation in the continuum mechanics to model an aggregate of independent molecules, which, although they can move freely, interact strongly with each other. The Navier-Stokes equations are enormously successful in modeling everything from atmospheric phenomena to producing practical engineering solutions, but in all these cases there is a number of molecules much larger than $10^{25}$ molecules. For sure $10^4$ water molecules together and interacting do not exhibit liquid behavior, but something different, while $10^{18}$ would have liquid behavior.

Answer 1

Whether the Navier-Stokes are or not a valid model is dictated by the Knudsen number:

$$Kn=\frac{\lambda}{L}$$

Where $\Lambda$ is the mean free path of the fluid and $L$ is a characteristic length scale of the problem. Usually the fluid can be modeled as a continuum if $Kn\leq 0.1$ (Lets remember that the Navier-Stokes equations can be obtained as a first order approximation from the Boltzmann equation by means of the Chapman-Enskog method, the higher order approximations are known as Burnett equations). Now, if the particles have a cross section $\sigma$ and density $n=N/V=N/L^3$ (we are considering a fluid in a box of length $L$), the mean free path can be (roughly) estimated as $\lambda=1/(\sigma n)=L^3/(\sigma N)$. If $Kn=0.1$ then $\lambda/L=L^2/(\sigma N)=0.1$. Then we can estimate the number as particles for which (above this value) the N-S equations are valid as $N=10\frac{L^2}{\sigma}$ or $\frac{L^2}{\sigma} \leq N$.

Answer 2

The Knudsen number is really only useful in dilute gases, where molecules typically fly a long way between collisions. For a dense fluids such as water, this is almost certainly not a good measure of the continuum equations being valid. Generally the validity of the continuum would be application specific. One way to test this is to run molecular dynamics (MD) similation and when the continuum and molecular models give the same results, assume the continuum is "valid".

For dense fluids in a channel, MD simulation suggest that fewer than 10 molecular diameters are needed to reproduce continuum behaviour. This would be a very small system. Below this size molecular-stacking behaviour near the wall explains the difference between the models. This could be argued to be a failure of the no-slip assumption usually required to close continuum models.

The number of molecules is important in the measured Noise. Molecular dynamics needs to be averaged in space and time to give a field which can be compared to the continuum. This field will still include fluctuation but these should be small enough to assume the continuum is valid (one estimate of this is given here). Otherwise a fluctuating hydrodynamics approach might be needed, where the Navier Stokes is extended to include a stoichastic term. Note that fewer molecules, averaged for longer times, might give better agreement with the continuum. In this case, a specific number of molecules to match the conitnuum is less useful than considering statistics.

More generally, the differences between the two models is less about the number of molecules used but instead failure of assumptions such as the no slip, constant viscosity, stress tensor symmetry, homogenous fluid (e.g. in multi-phase flows), etc. For more complex situations such as multi-phase flow, phase change and boiling, visoelastic fluids, extreme heating or shearing, chemical reactions, etc it is less clear if the continuum assumption itself is violated or if the continous equations used are not sufficiently complex to model the problem.

The work of Gad El Hak is very good on some of these discussions. Personally I believe using the weak or control volume form of the continuum equations solves some of these problems, as these can be shown to be valid for a single molecule.

Answer 3

After some thought, perhaps the question is not so much the total number of particles $N$, but the type of motion. In continuum mechanics, derivatives are used, so it is assumed that the field of velocities or displacements is a continuous function. Namely, it is assumed that there is a differentiable map $\phi:\mathcal{B}\times\mathbb{R} \to \mathbb{R}^3$, the position of the material point $\boldsymbol{X}$ will be $\boldsymbol{x}_t = \phi(\boldsymbol{X},t) = \phi_t(\boldsymbol{X})$ at time $t$. This implies for example that the velocity field exists:

$$\boldsymbol{V}_t = \frac{\partial \phi_t}{\partial t} $$

Now consider an orthogonal lattice of points $P_1,\dots, P_n$. And let us imagine that under the motion such a grid is transformed into a slightly deformed grid where we also admit scrambling or of points. Then it will no longer be possible to adequately define the motion as a continuous function, let alone differentiable. In order that such scrambling of points does not occur, it is necessary that the motions of the particles satisfy certain restrictions with respect to that of their neighbors.

It is clear that at the macroscopic scale we can "ignore" a handful of "wayward" particles that do not respect order and intrude between the trajectories of others, as long as the motion of the majority of particles is orderly. It is clear that with statistically few particles it becomes more likely that it is impossible to consistently represent the positions as continuous functions that interpolate the position of the majority of particles. Seen in this way, it should define the probability of of geometrically inconsistent scrambling/arrangements.