To understand lattice Boltzmann model capability for capturing non-equilibrium effects, the model with first-order
expansion of the equilibrium distribution function is analytically
investigated. In particular, the velocity profile of Couette flows is exactly
obtained for the D2Q9 model, which shows retaining the first order
expansion can capture rarefaction effects in the incompressible
limit. Meanwhile, it clearly demonstrates that the D2Q9 model is not
able to reflect flow characteristics in the Knudsen layer.