Nonlinear Poiseuille flow in a gas

Abstract

The nonlinear Boltzmann equation for the steady planar Poiseuille flow generated by an external field g is exactly solved through order g2. It is shown that the pressure and temperature profiles, as well as the momentum and heat fluxes, are in qualitative disagreement with the Navier–Stokes predictions. For instance, the temperature has a local minimum at the middle layer instead of a maximum. Also, a longitudinal component of the heat flux exists in the absence of gradients along that direction and normal stress differences appear although the flow is incompressible. To account for these g2-order effects, which are relevant when the hydrodynamic quantities change over a characteristic length of the order of the mean free path, it is shown that the Chapman–Enskog expansion should be carried out three steps beyond the Navier–Stokes level.