8.1 Overview

The Chapman-Enskog expansion of the velocity distribution in terms of the power series of the Knudsen number gives rise to various forms of approximation for the Boltzmann equation. As was shown in Chap. 4, for the thermal equilibrium condition, the Euler equations are obtained. As the departure from equilibrium increases, measured by appropriately defined Knudsen numbers, higher order terms need to be included. The first-order expansion results in the Navier-Stokes equations and the Burnet equations have also been derived as the second-order approximation form of the Boltzmann equation. For micro flows of small Knudsen number, say, less than 0.01, the Navier-Stokes equations have generally been found satisfactory. At higher Knudsen number, the Burnett equations become more appropriate. The application of the slip boundary condition for the velocity and the temperature jump condition often enables both the Navier-Stokes equations and the Burnett equations to provide solutions at high Knudsen numbers. At higher Knudsen number, it becomes necessary to use discrete based approaches, such as molecular dynamics (MD), direct simulation Monte Carlo (DSMC), or lattice Boltzmann method, for numerical simulations of practical flows. These first principle methods are physically sound and valid for flows at all Knudsen numbers. They, however, demand more computational resources than the differential equation models for flows of low speed and Knudsen number.

For microfluidic devices, the operational value of the Knudsen number can spread over quite a large range in the same system. A single-scale approach based on a continuum equation model is then not...