Resolving Knudsen Layer by High Order Moment Expansion

TL;DR

This paper introduces a high-order hyperbolic moment system to model the Knudsen layer in Kramers' problem, providing analytical solutions that improve accuracy over a range of conditions and approach the kinetic equation solution as the order increases.

Contribution

The paper develops an analytical high-order hyperbolic moment system that accurately models the Knudsen layer and reduces boundary conditions from kinetic boundary conditions.

Findings

Solutions approach the linearized BGK kinetic equation with increasing order

Velocity profile in the Knudsen layer is captured with improved accuracy

Analytical solutions are derived for the moment system

Abstract

We model the Knudsen layer in Kramers' problem by linearized high order hyperbolic moment system. Due to the hyperbolicity, the boundary conditions of the moment system is properly reduced from the kinetic boundary condition. For Kramers' problem, we give the analytical solutions of moment systems. With the order increasing of the moment model, the solutions are approaching to the solution of the linearized BGK kinetic equation. The velocity profile in the Knudsen layer is captured with improved accuracy for a wide range of accommodation coefficients.