In an effort to understand gas flow in a thin (nanometers) channel we have
investigated the Broadwell model. 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 interparticle collisions,
we are able to prove that under a diffusive scaling the solutions of
the full Broadwell model converge weakly to solutions of the
diffusion equation. Under a hyperbolic scaling, we are able to
prove 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 diffusive and hyperbolic scalings
must be weak. Joint work with Andrew Christlieb and James
Rossmanith.
|