The Broadwell model in a thin channel

Abstract

In this paper we are concerned with the limiting behavior of gas flow in a thin channel as described by the Broadwell model. The Broadwell model is a simplified kinetic description for gas dynamics where the main assumption is that the particle distribution function can be represented by a discrete number of velocities. Starting from the Broadwell model and the appropriate boundary conditions we derive two 1D models for gas transport in a thin channel. In the limit of no interparticle collisions the 1D model is the well known telegraph equation. In the case of collisional flow the 1D model is a system of three first-order hyperbolic PDEs. Both 1D models are validated through numerical simulations that compare the 1D models to the 2D Broadwell system. Furthermore, in the limit of no inter-particle collisions we are able to rigorously show that under a di.usive scaling the solutions of the full Broadwell model converge weakly to solutions of the diffusion equation. Under a hyperbolic scaling we are able to show that solutions to the collisionless Broadwell model converge weakly to the solutions of the telegraph equation. Finally, we derive a long-time asymptotic formula for the solution of the collisionless Broadwell system, which reveals oscillations that explain why the convergence in the di.usive and hyperbolic scalings must be weak. Due to the nonlinearity of the inter-particle collisions, we are not able to prove rigorous convergence results for the collisional Broadwell system.