An Approximate Analytical Solution to Knudsen Layers

TL;DR

This paper develops an approximate analytical solution for Knudsen layers using hyperbolic moment methods, effectively capturing temperature jumps with few moments and demonstrating convergence as more moments are added.

Contribution

It introduces a linearized hyperbolic moment system for Knudsen layers based on the Boltzmann equation with the Shakhov model, showing well-posedness and convergence.

Findings

System captures temperature jump coefficient effectively.

Few moments suffice for accurate thermal Knudsen layer modeling.

Qualitative convergence observed with increasing moments.

Abstract

We apply moment methods to obtaining an approximate analytical solution to Knudsen layers. Based on the hyperbolic regularized moment system for the Boltzmann equation with the Shakhov collision model, we derive a linearized hyperbolic moment system to model the scenario with the Knudsen layer vicinity to a solid wall with Maxwell boundary condition. We find that the reduced system is in an even-odd parity form that the reduced system proves to be well-posed under all accommodation coefficients. We show that the system may capture the temperature jump coefficient and the thermal Knudsen layer well with only a few moments. With the increasing number of moments used, qualitative convergence of the approximate solution is observed.