Chapman–Enskog theory

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 ${\displaystyle \tau _{c}}$ and the mean free time between collisions ${\displaystyle \tau _{f}}$: ${\displaystyle \tau _{c}\ll \tau _{f}}$. This condition ensures that collisions are well-defined events in space and time, and holds if the dimensionless parameter ${\displaystyle \gamma \equiv r_{c}^{3}n}$ is small, where ${\displaystyle r_{c}}$ 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 ${\displaystyle \tau _{f}}$ 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 smaller 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 do 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 ${\displaystyle \tau _{f}\ll \{\tau _{\text{ext}}\}}$. 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 to the Burnett equations and super-Burnett equations.

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 parameter ${\displaystyle \epsilon }$ is introduced to keep track of the appropriate orders in the Chapman–Enskog expansion:

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

It can be seen that small ${\displaystyle \epsilon }$ 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

${\displaystyle f=f^{(0)}+\epsilon f^{(1)}+\epsilon ^{2}f^{(2)}+\ldots \ .}$

Solutions which can be formally expanded in this way are known as normal solutions to the Boltzmann equation.[6] Clearly, this class of solutions excludes non-perturbative contributions (such as ${\displaystyle e^{-1/\epsilon }}$), which appear in boundary layers or near internal shock layers. Thus, Chapman–Enskog theory is restricted to situations in which such solutions are negligible.

Substituting this expansion and equating orders of ${\displaystyle \epsilon }$ leads to the hierarchy

${\displaystyle {\begin{aligned}J(f^{(0)},f^{(0)})&=0\\2J(f^{(0)},f^{(n)})&=\left({\frac {\partial }{\partial t}}+\mathbf {v\cdot } {\frac {\partial }{\partial \mathbf {r} }}+{\frac {\mathbf {F} }{m}}\cdot {\frac {\partial }{\partial \mathbf {v} }}\right)f^{(n-1)}-\sum _{m=1}^{n-1}J(f^{(n)},f^{(n-m)}),\qquad n>0,\end{aligned}}}$

where ${\displaystyle J}$ is an integral operator, linear in both its arguments, which satisfies ${\displaystyle J(f,g)=J(g,f)}$ and ${\displaystyle J(f,f)={\hat {C}}f}$. The solution to the first equation is a Gaussian:

${\displaystyle f^{(0)}=n'(\mathbf {r} ,t)\left({\frac {m}{2\pi k_{B}T'(\mathbf {r} ,t)}}\right)^{3/2}\exp \left[-{\frac {m\left(\mathbf {v} -\mathbf {v} '_{0}(\mathbf {r} ,t)\right)^{2}}{2k_{B}T'(\mathbf {r} ,t)}}\right].}$

for some functions ${\displaystyle n'(\mathbf {r} ,t)}$, ${\displaystyle \mathbf {v} '_{0}(\mathbf {r} ,t)}$, and ${\displaystyle T'(\mathbf {r} ,t)}$. It is tempting to equate these functions with the physical hydrodynamic fields defined as moments of ${\displaystyle f(\mathbf {r} ,\mathbf {v} ,t)}$:

${\displaystyle {\begin{aligned}n(\mathbf {r} ,t)&=\int fd\mathbf {v} \\n(\mathbf {r} ,t)v_{0}(\mathbf {r} ,t)&=\int \mathbf {v} fd\mathbf {v} \\n(\mathbf {r} ,t)T(\mathbf {r} ,t)&=\int {\frac {m}{3k_{B}}}\mathbf {v} ^{2}fd\mathbf {v} .\end{aligned}}}$

From a purely mathematical point of view, however, the two sets of functions are not necessarily the same for ${\displaystyle \epsilon >0}$ (for ${\displaystyle \epsilon =0}$ they are equal by definition). Indeed, proceeding systematically in the hierarchy, one finds that similarly to ${\displaystyle f^{(0)}}$, each ${\displaystyle f^{(n)}}$ also contains arbitrary functions of ${\displaystyle \mathbf {r} }$ and ${\displaystyle t}$ whose relation to the physical hydrodynamic fields is a priori unknown. One of the key simplifying assumptions of Chapman–Enskog theory is to assume that these otherwise arbitrary functions can be written in terms of the exact hydrodynamic fields and their spatial gradients. In other words, the space and time dependence of ${\displaystyle f}$ enters only implicitly through the hydrodynamic fields. This statement is physically plausible, since for small Knudsen numbers, one expects to enter the hydrodynamic regime in which the state of the gas is determined solely by the hydrodynamic fields. In the case of ${\displaystyle f^{(0)}}$, the functions ${\displaystyle n'(\mathbf {r} ,t)}$, ${\displaystyle \mathbf {v} '_{0}(\mathbf {r} ,t)}$, and ${\displaystyle T'(\mathbf {r} ,t)}$ are assumed exactly equal to the physical hydrodynamic fields.

While these assumptions are physically plausible, there is the question of whether solutions which satisfy these properties actually exist. More precisely, one must show that solutions exist satisfying

${\displaystyle {\begin{aligned}\int \sum _{n=1}^{\infty }\epsilon ^{n}f^{(n)}d\mathbf {v} =0=\int \sum _{n=1}^{\infty }\epsilon ^{n}f^{(n)}\mathbf {v} ^{2}d\mathbf {v} \\\int \sum _{n=1}^{\infty }\epsilon ^{n}f^{(n)}v_{i}d\mathbf {v} =0,\qquad i\in \{x,y,z\}.\end{aligned}}}$

Moreover, even if such solutions exist, there remains the additional question of whether they span the complete set of normal solutions to the Boltzmann equation, i.e. do not represent an artificial restriction of the original expansion in ${\displaystyle \epsilon }$. One of the key technical achievements of Chapman–Enskog theory is to answer both of these questions in the positive.[6] Thus, at least at the formal level, there is no loss of generality in the Chapman–Enskog approach.

With these formal considerations established, one can proceed to calculate ${\displaystyle f^{(1)}}$. The result is[1]

${\displaystyle f^{(1)}=\left[-{\frac {1}{n}}\left({\frac {2k_{B}T}{m}}\right)^{1/2}\mathbf {A(\mathbf {v} )\cdot \nabla \ln } T\mathbf {-} {\frac {2}{n}}\mathbb {B(\mathbf {v} )\colon \nabla } \mathbf {v} _{0}\right]f^{(0)},}$

where ${\displaystyle \mathbf {A} (\mathbf {v} )}$ is a vector and ${\displaystyle \mathbb {B} (\mathbf {v} )}$ a tensor, each a solution of a linear inhomogeneous integral equation that can be solved explicitly by a polynomial expansion. Note that the colon denotes the double dot product, ${\displaystyle \mathbb {T}$ :\mathbb {T'} =\sum _{i}\sum _{j}T_{ij}T'_{ji}} for tensors ${\displaystyle \mathbb {T} }$, ${\displaystyle \mathbb {T'} }$.

To first order in the Knudsen number, the heat flux ${\displaystyle \mathbf {q} ={\frac {m}{2}}\int f\mathbf {v} ^{2}\mathbf {v} d\mathbf {v} }$ is found to obey Fourier's law of heat conduction,[7]

${\displaystyle \mathbf {q} =-\lambda \nabla T,}$

and the momentum-flux tensor ${\displaystyle \mathbf {\sigma } =m\int (\mathbf {v} -\mathbf {v} _{0})(\mathbf {v} -\mathbf {v} _{0})^{\mathsf {T}}f\mathrm {d} \mathbf {v} }$ is that of a Newtonian fluid,[7]

${\displaystyle \mathbf {\sigma } =p\mathbb {I} -\mu \left(\nabla \mathbf {v_{0}} +\nabla \mathbf {v_{0}} ^{T}\right)+{\frac {2}{3}}\mu (\nabla \cdot \mathbf {v_{0}} )\mathbb {I} ,}$

with ${\displaystyle \mathbb {I} }$ the identity tensor. Here ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ are constants which we now identify with the thermal conductivity and viscosity. They can be calculated explicitly in terms of molecular parameters by solving a linear integral equation; the table below summarizes the results for a few important molecular models (${\displaystyle m}$ is the molecule mass and ${\displaystyle k_{B}}$ is the Boltzmann constant).[8]

Table 1: Predicted expressions for thermal conductivity and viscosity.

Model	${\displaystyle \mu }$	${\displaystyle \lambda }$	Notes
Rigid elastic spheres of diameter ${\displaystyle \sigma }$	${\displaystyle 1.016\cdot {\frac {5}{16\sigma ^{2}}}\left({\frac {k_{B}mT}{\pi }}\right)^{1/2}}$	${\displaystyle 2.522\cdot {\frac {3}{2}}{\frac {k_{B}}{m}}\cdot \mu }$	Correct to 3 decimal places.
Molecules with repulsive force ${\displaystyle \kappa /r^{\nu }}$	${\displaystyle {\frac {5}{8}}{\frac {1}{A_{2}(\nu )\Gamma \left(4-{\frac {2}{\nu -1}}\right)}}\left({\frac {k_{B}mT}{\pi }}\right)^{1/2}\left({\frac {2k_{B}T}{\kappa }}\right)^{2/(\nu -1)}}$	${\displaystyle {\frac {15}{4}}{\frac {k_{B}}{m}}\cdot \mu }$	${\displaystyle \Gamma }$ denotes the Gamma function, and ${\displaystyle A_{2}(\nu )}$ is a numerical factor. Chapman and Cowling list several values of the latter, e.g. ${\displaystyle A_{2}(5)=0.436}$ and ${\displaystyle A_{2}(11)=0.319}$.[9]
Lennard-Jones potential: ${\displaystyle V(r)=4\epsilon \{(\sigma /r)^{12}-(\sigma /r)^{6}\}}$	${\displaystyle {\frac {5}{8\sigma ^{2}}}\left({\frac {k_{B}mT}{\pi }}\right)^{1/2}\cdot {\frac {1}{{\mathcal {W}}_{1}^{(2)}(2)}}}$	${\displaystyle {\frac {15}{4}}{\frac {k_{B}}{m}}\cdot \mu }$	${\displaystyle {\mathcal {W}}_{1}^{(2)}(2)}$ is a function of ${\displaystyle k_{B}T/\epsilon }$ which can be calculated numerically. It varies from ${\displaystyle 5.682}$ for ${\displaystyle k_{B}T/\epsilon =0.3}$ to ${\displaystyle 1.1738}$ for ${\displaystyle k_{B}T/\epsilon =100}$.[10]

With these results, it is straightforward to obtain the Navier–Stokes equations. Taking velocity moments of the Boltzmann equation leads to the exact balance equations for the hydrodynamic fields ${\displaystyle n(\mathbf {r} ,t)}$, ${\displaystyle \mathbf {v} _{0}(\mathbf {r} ,t)}$, and ${\displaystyle T(\mathbf {r} ,t)}$:

${\displaystyle {\begin{aligned}{\frac {\partial n}{\partial t}}+\nabla \cdot \left(n\mathbf {v} _{0}\right)&=0\\{\frac {\partial \mathbf {v} _{0}}{\partial t}}+\mathbf {v} _{0}\cdot \nabla \mathbf {v} _{0}-{\frac {\mathbf {F} }{m}}+{\frac {1}{n}}\nabla \cdot \mathbf {\sigma } &=0\\{\frac {\partial T}{\partial t}}+\mathbf {v} _{0}\cdot \nabla T+{\frac {2}{3k_{B}n}}\left(\mathbf {\sigma$ :} \nabla \mathbf {v} _{0}+\nabla \cdot \mathbf {q} \right)&=0.\end{aligned}}}

As in the previous section the colon denotes the double dot product, ${\displaystyle \mathbb {T}$ :\mathbb {T'} =\sum _{i}\sum _{j}T_{ij}T'_{ji}} . Substituting the Chapman–Enskog expressions for ${\displaystyle \mathbf {q} }$ and ${\displaystyle \sigma }$, one arrives at the Navier–Stokes equations.

Comparison with experiment

An important prediction of Chapman–Enskog theory is that viscosity is independent of density (this can be seen for each molecular model in table 1, but is actually model-independent). This surprising result traces back to James Clerk Maxwell, who inferred it in 1860 on the basis of more elementary kinetic arguments.[11] It is well-verified experimentally for gases at ordinary densities.

Table 2: Experimentally measured values of ${\displaystyle f=\lambda /\mu c_{v}}$ for the first five noble gases.[12]

Helium	2.45
Neon	2.52
Argon	2.48
Krypton	2.535
Xenon	2.58

On the other hand, the theory predicts that ${\displaystyle \mu }$ does depend on temperature. For rigid elastic spheres, the predicted scaling is ${\displaystyle \mu \propto T^{1/2}}$, while other models typically show greater variation with temperature. For instance, for molecules repelling each other with force ${\displaystyle \propto r^{-\nu }}$ the predicted scaling is ${\displaystyle \mu \propto T^{s}}$, where ${\displaystyle s=1/2+2/(\nu -1)}$. Taking ${\displaystyle s=0.668}$, corresponding to ${\displaystyle \nu \approx 12.9}$, shows reasonable agreement with the experimentally observed scaling for helium. For more complex gases the agreement is not as good, most likely due to the neglect of attractive forces.[13] Indeed, the Lennard-Jones model, which does incorporate attractions, can be brought into closer agreement with experiment (albeit at the cost of a more opaque ${\displaystyle T}$ dependence; see the Lennard-Jones entry in table 1).[14]

Chapman–Enskog theory also predicts a simple relation between ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ in the form ${\displaystyle \lambda =f\mu c_{v}}$, where ${\displaystyle c_{v}}$ is the specific heat at constant volume and ${\displaystyle f}$ is a purely numerical factor. For spherically symmetric molecules, its value is predicted to be very close to ${\displaystyle 2.5}$ in a slightly model-dependent way. For instance, rigid elastic spheres have ${\displaystyle f\approx 2.522}$, and molecules with repulsive force ${\displaystyle \propto r^{-13}}$ have ${\displaystyle f\approx 2.511}$ (the latter deviation is ignored in table 1). The special case of Maxwell molecules (repulsive force ${\displaystyle \propto r^{-5}}$) has ${\displaystyle f=2.5}$ exactly.[15] Since ${\displaystyle \lambda }$, ${\displaystyle \mu }$, and ${\displaystyle c_{v}}$ can be measured directly in experiments, an easy experimental test of Chapman–Enskog theory is to measure ${\displaystyle f}$ for the spherically symmetric noble gases. Table 2 shows that there is reasonable agreement between theory and experiment.[12]

The basic principles of Chapman–Enskog theory can be extended to more diverse physical models, including gas mixtures and molecules with internal degrees of freedom. In the high-density regime, the theory can be adapted to account for collisional transport of momentum and energy, i.e. transport over a molecular diameter during a collision, rather than over a mean free path (in between collisions). Including this mechanism predicts a density dependence of the viscosity at high enough density, which is also observed experimentally.

One can also carry out the theory to higher order in the Knudsen number. In particular, the third-order contribution ${\displaystyle f^{(2)}}$ has been calculated by Burnett.[16] In general circumstances, however, these high-order corrections should be approached with caution, given that the Chapman–Enskog expansion may not always converge.[17] (On the other hand, the expansion is thought to be at least asymptotic to solutions of the Boltzmann equation, in which case truncating at low order still gives accurate results.)[18] Even if the higher order corrections do afford improvement in a given system, the interpretation of the corresponding hydrodynamical equations is still debated.[19]

• Transport phenomena
• Kinetic theory of gases
• Boltzmann equation
• Navier–Stokes equations
• Viscosity
• Thermal conductivity

The classic monograph on the topic:

• Chapman, Sydney; Cowling, T.G. (1970), The Mathematical Theory of Non-Uniform Gases (3rd ed.), Cambridge University Press

Contains a technical introduction to normal solutions of the Boltzmann equation:

• Grad, Harold (1958), "Principles of the Kinetic Theory of Gases", in Flügge, S. (ed.), Encyclopedia of Physics, XII, Springer-Verlag, pp. 205–294