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.

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.

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