Chapman–Enskog theory

Statistical mechanics framework

Chapman–Enskog theory provides a framework in which equations of hydrodynamics for a gas can be derived from the Boltzmann equation. The technique justifies the otherwise phenomenological constitutive relations appearing in hydrodynamical descriptions such as the Navier–Stokes equations. In doing so, expressions for various transport coefficients such as thermal conductivity and viscosity are obtained in terms of molecular parameters. Thus, Chapman–Enskog theory constitutes an important step in the passage from a microscopic, particle-based description to a continuum hydrodynamical one.

The theory is named for Sydney Chapman and David Enskog, who introduced it independently in 1916 and 1917.^[1]

Description

The starting point of Chapman–Enskog theory is the Boltzmann equation for the 1-particle distribution function ${\displaystyle f(\mathbf {r} ,\mathbf {v} ,t)}$:

${\displaystyle {\frac {\partial f}{\partial t}}+\mathbf {v\cdot } {\frac {\partial f}{\partial \mathbf {r} }}+{\frac {\mathbf {F} }{m}}\cdot {\frac {\partial f}{\partial \mathbf {v} }}={\hat {C}}f,}$

where ${\displaystyle {\hat {C}}}$ is a nonlinear integral operator which models the evolution of ${\displaystyle f}$ under interparticle collisions. This nonlinearity makes solving the full Boltzmann equation difficult, and motivates the development of approximate techniques such as the one provided by Chapman–Enskog theory.

Given this starting point, the various assumptions underlying the Boltzmann equation carry over to Chapman–Enskog theory as well. The most basic of these requires a separation of scale between the collision duration ${\displaystyle \tau _{\mathrm {c} }}$ and the mean free time between collisions ${\displaystyle \tau _{\mathrm {f} }}$: ${\displaystyle \tau _{\mathrm {c} }\ll \tau _{\mathrm {f} }}$. This condition ensures that collisions are well-defined events in space and time, and holds if the dimensionless parameter ${\displaystyle \gamma \equiv r_{\mathrm {c} }^{3}n}$ is small, where ${\displaystyle r_{\mathrm {c} }}$ is the range of interparticle interactions and ${\displaystyle n}$ is the number density.^[2] In addition to this assumption, Chapman–Enskog theory also requires that ${\displaystyle \tau _{\mathrm {f} }}$ is much smaller than any extrinsic timescales ${\displaystyle \tau _{\text{ext}}}$. These are the timescales associated with the terms on the left hand side of the Boltzmann equation, which describe variations of the gas state over macroscopic lengths. Typically, their values are determined by initial/boundary conditions and/or external fields. This separation of scales implies that the collisional term on the right hand side of the Boltzmann equation is much larger than the streaming terms on the left hand side. Thus, an approximate solution can be found from

${\displaystyle {\hat {C}}f=0.}$

It can be shown that the solution to this equation is a Gaussian:

${\displaystyle f=n(\mathbf {r} ,t)\left({\frac {m}{2\pi k_{B}T(\mathbf {r} ,t)}}\right)^{3/2}\exp \left[-{\frac {m\left(\mathbf {v} -\mathbf {v} _{0}(\mathbf {r} ,t)\right)^{2}}{2k_{B}T(\mathbf {r} ,t)}}\right],}$

where ${\displaystyle m}$ is the molecule mass and ${\displaystyle k_{B}}$ is the Boltzmann constant.^[3]A gas is said to be in local equilibrium if it satisfies this equation.^[4] The assumption of local equilibrium leads directly to the Euler equations, which describe fluids without dissipation, i.e. with thermal conductivity and viscosity equal to ${\displaystyle 0}$. The primary goal of Chapman–Enskog theory is to systematically obtain generalizations of the Euler equations which incorporate dissipation. This is achieved by expressing deviations from local equilibrium as a perturbative series in Knudsen number ${\displaystyle {\text{Kn}}}$, which is small if ${\displaystyle \tau _{\mathrm {f} }\ll \tau _{\text{ext}}}$. Conceptually, the resulting hydrodynamic equations describe the dynamical interplay between free streaming and interparticle collisions. The latter tend to drive the gas towards local equilibrium, while the former acts across spatial inhomogeneities to drive the gas away from local equilibrium.^[5] When the Knudsen number is of the order of 1 or greater, the gas in the system being considered cannot be described as a fluid.

To first order in ${\displaystyle {\text{Kn}}}$ one obtains the Navier–Stokes equations. Second and third orders give rise, respectively, to the Burnett equations and super-Burnett equations.

Mathematical formulation

Since the Knudsen number does not appear explicitly in the Boltzmann equation, but rather implicitly in terms of the distribution function and boundary conditions, a dummy variable ${\displaystyle \varepsilon }$ is introduced to keep track of the appropriate orders in the Chapman–Enskog expansion:

${\displaystyle {\frac {\partial f}{\partial t}}+\mathbf {v\cdot } {\frac {\partial f}{\partial \mathbf {r} }}+{\frac {\mathbf {F} }{m}}\cdot {\frac {\partial f}{\partial \mathbf {v} }}={\frac {1}{\varepsilon }}{\hat {C}}f.}$

Small ${\displaystyle \varepsilon }$ implies the collisional term ${\displaystyle {\hat {C}}f}$ dominates the streaming term ${\displaystyle \mathbf {v\cdot } {\frac {\partial f}{\partial \mathbf {r} }}+{\frac {\mathbf {F} }{m}}\cdot {\frac {\partial f}{\partial \mathbf {v} }}}$, which is the same as saying the Knudsen number is small. Thus, the appropriate form for the Chapman–Enskog expansion is

${\displaystyle f=f^{(0)}+\varepsilon f^{(1)}+\varepsilon ^{2}f^{(2)}+\cdots \ .}$

Solutions that can be formally expanded in this way are known as normal solutions to the Boltzmann equation.^[6] This class of solutions excludes non-perturbative contributions (such as ${\displaystyle e^{-1/\varepsilon }}$), which appear in boundary layers or near internal