Linear Moment Models to Approximate Knudsen Layers

TL;DR

This paper introduces a new Maxwell-type boundary condition for linear moment systems to accurately model Knudsen layers near walls, ensuring well-posedness and high accuracy with few moments.

Contribution

It proposes a novel boundary condition for linear moment equations, guaranteeing well-posedness and effective modeling of Knudsen layers.

Findings

The boundary condition ensures unique solutions for half-space problems.

The model accurately captures Knudsen layers with minimal moments.

Application to classical flow problems demonstrates high precision.

Abstract

We propose a well-posed Maxwell-type boundary condition for the linear moment system in half-space. As a reduction of the Boltzmann equation, the moment equations are available to model Knudsen layers near a solid wall, where proper boundary conditions should be prescribed. Utilizing an orthogonal decomposition, we separate the part with a damping term from the system and then impose a new class of Maxwell-type boundary conditions on it. Due to the new boundary condition, we show that the half-space boundary value problem admits a unique solution with explicit expressions. Instantly, the well-posedness of the linear moment system is achieved. We apply the procedure to classical flow problems with the Shakhov collision term, such as the velocity slip and temperature jump problems. The model can capture Knudsen layers with very high accuracy using only a few moments.