Analyse du flux de Poiseuille bidimensionnel via l'équation de Boltzmann

Abstract

The two-dimensional Poiseuille flow induced by an external force is analysed in the framework of Boltzmann–Maxwell kinetic theory. In the limit of small Knudsen numbers (Kn [Formula: see text] 0.1), Boltzmann's nonlinear equation, written in terms of moments, is solved using perturbation theory. In our results, the hydrodynamic variable profiles are determined up to the fourth order in the perturbation parameter. Nonetheless, the method of solution remains valid to obtain all physical quantities of a gas undergoing Poiseuille flow. The major conclusion of our analysis has two elements. First, the profiles of hydrodynamic variables in two dimensions differ quantitatively (and sometimes qualitatively) from those in plane geometry. Thus, the Poiseuille flow representation in a cylindrical pipe is more accurate than in a canal between two parallel planes. Second, a critical comparaison between the theoretical predictions of the kinetic theory and those of Navier–Stokes shows that the two theories agree only up to the first order of perturbation. Starting at the second order, the difference between the two increases. Thus, within the limit of validity of the present study, the description by Navier–Stokes remains insufficient to predict the correct profiles for the hydrodynamic variables in the Poisseuille flow. [Journal translation]