Asymptotic expansion and Padé approximants for acceleration-driven Poiseuille flow of a rarefied gas: Bulk hydrodynamics and rheology.

Abstract

The perturbation expansion technique is employed to solve the Boltzmann equation for the acceleration-driven steady Poiseuille flow of a dilute molecular gas flowing through a planar channel. Neglecting wall effects and focusing only on the bulk hydrodynamics and rheology, the perturbation solution is sought around the channel centerline in powers of the strength of acceleration. To make analytical progress, the collision term has been approximated by the Bhatnagar-Gross-Krook kinetic model for hard spheres, and the related problem for Maxwell molecules was analyzed previously by Tij and Santos [J. Stat. Phys. 76, 1399 (1994)JSTPBS0022-471510.1007/BF02187068]. The analytical expressions for hydrodynamic (velocity, temperature, and pressure) and rheological fields (normal stress differences, shear viscosity, and heat flux) are obtained by retaining terms up to tenth order in acceleration, with one aim of the present work being to understand the convergence properties of the underlying perturbation series solutions. In addition, various rarefaction effects (e.g., the bimodal shape of the temperature profile, nonuniform pressure profile, normal-stress differences, and tangential heat flux) are also critically analyzed in the Poiseuille flow as functions of the local Froude number. The hydrodynamic and rheological fields evaluated at the channel centerline confirmed the oscillatory nature of the present series solutions (when terms of increasing order are sequentially included), signaling the well-known pitfalls of asymptotic expansion. The Padé approximation technique is subsequently applied to check the region of convergence of each series solution. It is found that the diagonal Padé approximants for rheological fields agree qualitatively with previous simulation data on acceleration-driven rarefied Poiseuille flow.