Chapman–Enskog theory

Chapman–Enskog theory presents accurate formulas for a multicomponent gas mixture under thermal and chemical equilibrium. In elastic gases the deviation from the Maxwell–Boltzmann distribution in the equilibrium is small and it can be treated as a perturbation. This method was aimed to obtain transport equations more general than the Euler equations. It is named for Sydney Chapman and David Enskog.

Chapman–Enskog Expansion

Solutions to the Navier–Stokes equations can be used to describe many fluid-dynamical phenomena such as laminar flows, turbulence and solitons. Fundamentally, the Navier-Stokes equation is derived from the Boltzmann equation. If particular models of the microscopic collision process are applied, explicit formulas for the transport equations can be acquired. The term Chapman-Enskog Expansion denotes this derivation of the Navier-Stokes equation and its transport coefficients from the Boltzmann equation and certain microscopic collision models. It was introduced independently by Chapman and Enskog between 1910 and 1920.

The expansion parameter of Chapman-Enskog is the Knudsen number, Kn. When it is of the order of 1 or greater, the gas in the system being considered cannot be described as a fluid. Also, the series produced from the Chapman-Enskog method is likely not to be convergent but asymptotic. This is implied by the application to the dispersion of sound. With higher order approximations of the Chapman-Enskog method, the Burnett and super-Burnett equations are attained, which have never been applied systematically. A complication with these equations is the subject of appropriate boundary conditions.

Mathematical Formulation

The classical, collisional Boltzmann equation typically has the following form

$\frac{\partial f}{\partial t} + \mathbf{v}\cdot\frac{\partial f}{\partial \mathbf{x}} + \mathbf{F} \cdot \frac{\partial f}{\partial \mathbf{p}} = B\left( f \otimes f \right)$

Where f is the probability density function, t is the time, x is the position, v the velocity, p the momentum, F the force and B represents the binary collision term. The collision term is usually some correlation function of f, so it involves an integral where f appears twice in the integrand, albeit at different positions or momenta. The collision term is nonlinear, which makes it difficult to solve this equation. The Chapman-Enskog theory is a way to linearize the Boltzmann equation using a perturbation expansion for f for some small parameter $\varepsilon$:

$f = f_0 + \varepsilon f_1 + \varepsilon^2 f_2 + ... = \sum_{n=0}^{\infty} \varepsilon^n f_n$

Given some initial unperturbed distribution function $f_0$, higher orders can be obtained by solving a linear integro - differential equation that depends on lower orders:

$\frac{\partial f_n}{\partial t} + \mathbf{v}\cdot\frac{\partial f_n}{\partial \mathbf{x}} + \mathbf{F} \cdot \frac{\partial f_n}{\partial \mathbf{p}} = \sum_{i=0}^n B\left( f_i \otimes f_{n-i} \right)$

See also

• Thermodiffusion

Bibliography

• Sydney Chapman; Thomas George Cowling, The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases, Cambridge University Press, 1990. ISBN 0-521-40844-X
• Dieter A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction, Springer, 2000. ISBN 3540669736
• Richard S. Ellis, , Chapman-Enskog-Hilbert Expansion for a Markovian Model of the Boltzmann equation, Communications on Pure an Applied Mathematics, vol. XXLVI, 327-359 (1973)