Diffusive Boltzmann equation, its fluid dynamics, Couette flow and Knudsen layers

TL;DR

This paper introduces a modified Boltzmann equation with a diffusive term derived from a stochastic process, and investigates its implications for fluid dynamics, boundary layers, and heat flux in rarefied gas flows.

Contribution

It develops a new stochastic process leading to a diffusive Boltzmann equation and explores its fluid dynamic consequences, including boundary layers and heat flux behavior.

Findings

Knudsen velocity boundary layers form with various closures.

Parallel heat flux near walls is significant and well-approximated by Grad closure.

Diffusive terms influence boundary layer and heat flux characteristics.

Abstract

In the current work we construct a multimolecule random process which leads to the Boltzmann equation in the appropriate limit, and which is different from the deterministic real gas dynamics process. We approximate the statistical difference between the two processes via a suitable diffusion process, which is obtained in the multiscale homogenization limit. The resulting Boltzmann equation acquires a new spatially diffusive term, which subsequently manifests in the corresponding fluid dynamics equations. We test the Navier-Stokes and Grad closures of the diffusive fluid dynamics equations in the numerical experiments with the Couette flow for argon and nitrogen, and compare the results with the corresponding Direct Simulation Monte Carlo (DSMC) computations. We discover that the full-fledged Knudsen velocity boundary layers develop with all tested closures when the viscosity and diffusivity are appropriately scaled in the vicinity of the walls. Additionally, we find that the component of the heat flux parallel to the direction of the flow is comparable in magnitude to its transversal component near the walls, and that the nonequilibrium Grad closure approximates this parallel heat flux with good accuracy.