Maxwellian gas undergoing a stationary Poiseuille flow in a pipe

TL;DR

This paper solves the moment equations from the Boltzmann equation for Maxwell molecules in a stationary Poiseuille flow, revealing limitations of Navier-Stokes theory and assessing kinetic models' accuracy.

Contribution

It provides a perturbation expansion solution to the Boltzmann equation for Poiseuille flow and compares its predictions with Navier-Stokes and BGK models.

Findings

Navier-Stokes theory fails to accurately predict flow profiles beyond first order.

The BGK model captures the qualitative behavior but with quantitative deviations.

Non-monotonic temperature profiles and normal stress differences are observed.

Abstract

The hierarchy of moment equations derived from the nonlinear Boltzmann equation is solved for a gas of Maxwell molecules undergoing a stationary Poiseuille flow induced by an external force in a pipe. The solution is obtained as a perturbation expansion in powers of the force (through third order). A critical comparison is done between the Navier-Stokes theory and the predictions obtained from the Boltzmann equation for the profiles of the hydrodynamic quantities and their fluxes. The Navier-Stokes description fails to first order and, especially, to second order in the force. Thus, the hydrostatic pressure is not uniform, the temperature profile exhibits a non-monotonic behavior, a longitudinal component of the flux exists in the absence of longitudinal thermal gradient, and normal stress differences are present. On the other hand, comparison with the Bhatnagar-Gross-Krook model kinetic equation shows that the latter is able to capture the correct functional dependence of the fields, although the numerical values of the coefficients are in general between 0.38 and 1.38 times the Boltzmann values. A short comparison with the results corresponding to the planar Poiseuille flow is also carried out.