Knudsen number

In 1755, L. Euler derived from the basic Newton laws of mechanics an equation for the motion of a fluid without taking into account the viscosity (cf. also Euler equation). To introduce the viscosity, and also compressibility, this equation has to be modified in accordance with the work of L. Navier (1821) and G.C. Stokes (1842) (cf. also Navier–Stokes equations). Such an equation can be written in dimensionless form by introducing three dimensionless numbers: the Mach number, the Reynolds number and the Prandtl number.

The Mach number is the ratio of the mean velocity $u$ of the fluid to the velocity of sound $c$, the Reynolds number is the product of the characteristic length of the vessel $l$ with the mean velocity $u$ divided by the viscosity $\nu$, and the Prandlt number is the ratio of the viscosity and the thermometric conductivity $\kappa$:

\begin{equation*} \operatorname{Ma} = \frac { u } { c } , \operatorname{Re} = \frac { u l } { \nu } , \operatorname{Pr} = \frac { \nu } { \kappa }. \end{equation*}

The discovery of atoms and molecules and the understanding of the role they play in the "human size" world was finalized around the turn into the nineteenth century, starting with A. Avogadro in 1833, giving an order of magnitude of the number $N \simeq 10 ^ { 19 }$ of molecules per volume element, continuing with J.R. Clausius, who in 1858 introduced the mean free path, and culminating with the 1905 paper of A. Einstein on Brownian motion.

The connection between the macroscopic and the molecular point of view was one of the challenges of the end of nineteenth century and still (as of 2000) attracts the attention of physicists and mathematicians. It is important both for a full understanding of basic laws of physics, like the appearance of irreversibility, and for practical purpose. Computation cannot be done at the level of molecules because the number of degree of freedom of the system, of the order of $N ^ { 6 }$, is definitely too large.

On the other hand, several modern technological problems (the re-entry of a space shuttle in the atmosphere, the computation of the current in a semi-conductor which is so small that the gas of electrons cannot reach thermal equilibrium, or the analysis of the ionized air between the reading head and a compact disc) involve gases of particles which are too rarefied to be computed with the macroscopic laws of physics.

To bridge such a gap, an intermediate description, the kinetic description, was given by J.C. Maxwell (1860), who introduced a density function $f ( x , v , t )$ which describes the probability of having at time $t$ and at the point $x$ a particle with the velocity $v$, and by L. Boltzmann who eventually (1872) derived an equation for this density (cf. also Boltzmann equation).

Therefore, with the number of particles increasing one should deduce from the molecular description the kinetic description and from the kinetic the Navier–Stokes equations. However, in this process it is not the macroscopic density that matters, but the average number of collisions that a molecule of "radius" $\sigma$ experiences during a unit time. It is given by the formula:

\begin{equation*} n = \pi \sigma ^ { 2 } N c. \end{equation*}

The averaged time for collision is then $n ^ { - 1 }$ and the average distance travelled during this time, i.e. the mean free path, is

\begin{equation*} \lambda = n ^ { - 1 } c = ( \pi \sigma ^ { 2 } N ) ^ { - 1 }. \end{equation*}

Observe that the total volume occupied by the gas (i.e. the packed volume) is given by

\begin{equation*} \mathcal{V} = \frac { 4 } { 3 } \pi \sigma ^ { 2 } N, \end{equation*}

and therefore is in a regime where $N$ is large, of the order of $10 ^ { 19 }$ and $\lambda$ of the order of unity, the packed volume is of the order $10 ^ { - 8 }$; hence the limit

\begin{equation*} N \rightarrow \infty , \sigma \rightarrow 0 , \frac { 1 } { \lambda } = \operatorname { lim } ( \pi \sigma ^ { 2 } N ) \in ] 0 , \infty [ \end{equation*}

corresponds to a rarefied gas, described by the Boltzmann equation. Again, this equation is written in dimensionless form with the introduction of the Knudsen number, the ratio of the mean free path to the characteristic length of the vessel containing the gas:

\begin{equation*} Kn = \frac { \lambda } { l }. \end{equation*}

As M.H.Ch. Knudsen (1871–1949), professor of physic at the university of Copenhagen, discovered, it is rather this number than the mean free path itself that contains the decisive information. As said above, when the Knudsen number is very small, intermolecular collisions dominate and the kinetic approximation becomes valid; if this number is large (in this situation one refers to the Knudsen gas), the molecules evolve almost freely in the vessel and the effect of the collisions with the boundaries dominate.

Furthermore, writing

\begin{equation*} \frac { \text{Ma} } { \text{Re} } = \frac { u / c } { u l / \nu } = \frac { 1 } { c } \frac { \nu } { \lambda }, \end{equation*}

one observes that the expression $\nu / \lambda$ is independent of the gas motion and homogeneous to a speed; the only such speed can be the speed of sound $c$ (up to some unessential "pure" constant $\alpha$) and therefore one obtains the von Kármán relation

\begin{equation*} \operatorname{Kn} = \alpha \frac {\operatorname{ Ma} } {\operatorname{ Re} }, \end{equation*}

which is due to Th. von Kármán (1923) and which is of paramount importance for the next step of the discussion: the derivation of the macroscopic equations from the kinetic description.

This program was initialized by D. Hilbert (1916), who proved that when the Knudsen number goes to zero, a first-order macroscopic approximation is provided by the compressible Euler equation. The connection with the Navier–Stokes equation turned out to be more subtle. Independently, S. Chapman (1916) and D. Enskog (1917) proved that a second-order correction is provided by the compressible Navier–Stokes equation with Reynolds number and Mach number of the order of the Knudsen number (cf. also Chapman–Enskog method). This is in perfect agreement with the von Kármán relation. Even more, the Chapman–Enskog derivation may be used to provide a "mathematical" proof of the von Kármán relation.

Eventually, the von Kármán relation implies that it is only in the zero Mach limit, i.e. in the incompressible limit, that the Navier–Stokes equations with finite Reynolds number are relevant.

A detailed description of recent mathematical work on this subject can be found in [a1], [a2].

Passing through the kinetic equation (first having a finite Knudsen number and then letting only this number go to infinity) implies, with relaxation through Maxwellian distributions, that only perfect gases, incompressible or with a pressure law given by the relation

\begin{equation*} p = \rho R T, \end{equation*}

are described at the macroscopic level.

For other constitutive relations, in particular with pressure given by the van der Waals law, in spite of some preliminary work [a3], [a4], the mathematical theory is in the preliminary stage and the role of the Knudsen number is not evident.

References

[a1]	C. Cercignani, R. Illner, M. Pulvirenti, "The mathematical theory of dilute gases" , Applied Math. Sci. , Springer (1994)
[a2]	F. Bouchut, F. Golse, M. Pulvirenti, "Kinetic equations and asymptotic theories" , Series in Appl. Math. , 4 , Elsevier/Gauthier-Villars (2000)
[a3]	C. Cercignani, M. Lampis, "On the H-theorem for polyatomic gases" J. Statist. Phys. , 26 (1981) pp. 795–801
[a4]	C. Morrey, "On the derivation of the equations of hydrodynamics from statistical mechanics" Commun. Pure Appl. Math. , 8 (1955) pp. 279–326