Chapman-Enskog Closure

The classic example of an asymptotic closure scheme is the Chapman-Enskog theory of a neutral gas dominated by collisions. In this theory, the small parameter is the ratio of the mean-free-path between collisions to the macroscopic variation lengthscale. It is instructive to briefly examine this theory, which is very well described in a classic monograph by Chapman and Cowling (Chapman and Cowling 1953).

Consider a neutral gas consisting of identical hard-sphere molecules of mass $m$ and diameter $\sigma$. Admittedly, this is not a particularly physical model of a neutral gas, but we are only considering it for illustrative purposes. The fluid equations for such a gas are similar to Equations (4.47)–(4.49):

$$\frac{dn}{dt} + n\,\nabla\cdot{\bf V} =0,$$ (4.55)

$$m\, n\,\frac{d {\bf V}}{dt} + \nabla p + \nabla\cdot \mbox{\boldmath$\pi$} + m\,n\,{\bf g} ={\bf0},$$ (4.56)

$$\frac{3}{2}\frac{d p}{dt} + \frac{5}{2}\,p\,\nabla \cdot {\bf V} + \mbox{\boldmath$\pi$} :\nabla{\bf V} + \nabla\cdot{\bf q} = 0.$$ (4.57)

Here, $n$ is the particle number density, ${\bf V}$ the flow velocity, $p$ the scalar pressure, and ${\bf g}$ the acceleration due to gravity. We have dropped the subscript $s$ because, in this case, there is only a single species. There is no collisional friction or heating in a single species system. Of course, there are no electrical or magnetic forces in a neutral gas, so we have included gravitational forces instead. The purpose of the closure scheme is to express the viscosity tensor, $\pi$, and the heat flux density, ${\bf q}$, in terms of $n$, ${\bf V}$, or $p$, and, thereby, complete the set of equations.

The mean-free-path, $l$, for hard-sphere molecules is given by

$$l = \frac{1}{\sqrt{2}\,\pi\,n\,\sigma^2}.$$ (4.58)

This formula is fairly easy to understand. The volume swept out by a given molecule in moving a mean-free-path must contain, on average, approximately one other molecule. Observe that $l$ is completely independent of the speed or mass of the molecules. The mean-free-path is assumed to be much smaller than the variation lengthscale, $L$, of macroscopic quantities, so that

$$\epsilon = \frac{l}{L} \ll 1.$$ (4.59)

In the Chapman-Enskog scheme, the distribution function is expanded, order by order, in the small parameter $\epsilon$:

$$f({\bf r}, {\bf v}, t) = f_0({\bf r}, {\bf v}, t) + \epsilon\,f_1({\bf r}, {\bf v}, t) + \epsilon^2\, f_2({\bf r}, {\bf v}, t) + \cdots.$$ (4.60)

Here, $f_0$, $f_1$, $f_2$, and so on, are all assumed to be of the same order of magnitude. In fact, only the first two terms in this expansion are ever calculated. To zeroth order in $\epsilon$, the kinetic equation requires that $f_0$ be a Maxwellian:

$$f_0({\bf r}, {\bf v}, t) = n({\bf r}) \left[\frac{m}{2\pi\,T({\bf... ...ght]^{3/2}\,\exp\!\left[-\frac{m\,({\bf v}- {\bf V})^2} {2\,T({\bf r})}\right].$$ (4.61)

Recall that $p=n\,T$. As is well known, there is zero heat flow or viscous stress associated with a Maxwellian distribution function (Reif 1965). Thus, both the heat flux density, ${\bf q}$, and the viscosity tensor, $\pi$, depend on the first-order non-Maxwellian correction to the distribution function, $f_1$.

It is possible to linearize the kinetic equation, and then rearrange it so as to obtain an integral equation for $f_1$ in terms of $f_0$. This rearrangement crucially depends on the bilinearity of the collision operator. Incidentally, the equation is integral because the collision operator is an integral operator. The integral equation is solved by expanding $f_1$ in velocity space using Laguerre polynomials (sometimes called Sonine polynomials) (Abramowitz and Stegun 1965). It is possible to reduce the integral equation to an infinite set of simultaneous algebraic equations for the coefficients in this expansion. If the expansion is truncated, after $N$ terms, say, then these algebraic equations can be solved for the coefficients. It turns out that the Laguerre polynomial expansion converges very rapidly. Thus, it is conventional to keep only the first two terms in this expansion, which is usually sufficient to ensure an accuracy of about $1$ percent in the final result. Finally, the appropriate moments of $f_1$ are taken, so as to obtain expression for the heat flux density and the viscosity tensor. Strictly speaking, after evaluating $f_1$, we should then go on to evaluate $f_2$, so as to ensure that $f_2$ really is negligible compared to $f_1$. In reality, this is never done because the mathematical difficulties involved in such a calculation are prohibitive.

The Chapman-Enskog method outlined previously can be applied to any assumed force law between molecules, provided that the force is sufficiently short-range (i.e., provided that it falls off faster with increasing separation than the Coulomb force). For all sensible force laws, the viscosity tensor is given by

$$\pi_{\alpha\beta} =- \eta \left( \frac{\partial V_\alpha}{\partia... ...ial r_\alpha} - \frac{2}{3}\,\nabla\cdot {\bf V}\,\delta_{\alpha\beta} \right),$$ (4.62)

whereas the heat flux density takes the form

$${\bf q} = - \kappa\,\nabla T.$$ (4.63)

Here, $\eta$ is the coefficient of viscosity, and $\kappa$ is the coefficient of thermal conductivity. It is convenient to write

$$\eta =m \,n \,{\mit\chi}_v,$$ (4.64)

$$\kappa = n\,{\mit\chi}_t,$$ (4.65)

where ${\mit\chi}_v$ is the viscous diffusivity and ${\mit\chi}_t$ is the thermal diffusivity. Both ${\mit\chi}_v$ and ${\mit\chi}_t$ have the dimensions of length squared over time, and are, effectively, diffusion coefficients. For the special case of hard-sphere molecules, Chapman-Enskog theory yields (Chapman and Cowling 1953):

$${\mit\chi}_v = \frac{75\pi^{1/2}}{64}\left(1+ \frac{3}{202} +\cdots\right)\nu\,l^{2} = A_v\,\nu\,l^{2},$$ (4.66)

$${\mit\chi}_t = \frac{5\pi^{1/2}}{16}\left( 1+ \frac{1}{44}+\cdots\right)\nu\,l^{2} = A_t\,\nu\,l^{2}.$$ (4.67)

Here,

$$\nu = \frac{v_t}{l}$$ (4.68)

is the collision frequency, and

$$v_t = \sqrt{\frac{2\,T}{m}}$$ (4.69)

is the thermal velocity. The first two terms in the Laguerre polynomial expansion are shown explicitly (in the round brackets) in Equations (4.66) and (4.67).

Equations (4.66) and (4.67) have a simple physical interpretation. The viscous and thermal diffusivities of a neutral gas can be accounted for in terms of the random-walk diffusion of molecules with excess momentum and energy, respectively. Recall the standard result in stochastic theory that if particles jump an average distance $l$, in a random direction, $\nu$ times a second, then the diffusivity associated with such motion is $\chi\sim\nu\,l^{2}$ (Reif 1965). Chapman-Enskog theory basically allows us to calculate the numerical constants $A_v$ and $A_t$, multiplying $\nu\,l^{2}$ in the expressions for $\chi_v$ and $\chi_t$, for a given force law between molecules. Obviously, these coefficients are different for different force laws. The expression for the mean-free-path, $l$, is also different for different force laws.