An Extension to the Navier–Stokes Equations to Incorporate Gas Molecular Collisions With Boundaries

Abstract

We investigate a model for microgas-flows consisting of the Navier–Stokes equations extended to include a description of molecular collisions with solid-boundaries together with first- and second-order velocity-slip boundary conditions. By considering molecular collisions affected by boundaries in gas flows, we capture some of the near-wall effects that the conventional Navier–Stokes equations with a linear stress-/strain-rate relationship are unable to describe. Our model is expressed through a geometry-dependent mean-free-path yielding a new viscosity expression, which makes the stress-/strain-rate constitutive relationship nonlinear. Test cases consisting of Couette and Poiseuille flows are solved using these extended Navier–Stokes equations and we compare the resulting velocity profiles with conventional Navier–Stokes solutions and those from the BGK kinetic model. The Poiseuille mass flow rate results are compared with results from the BGK-model and experimental data for various degrees of rarefaction. We assess the range of applicability of our model and show that it can extend the applicability of conventional fluid dynamic techniques into the early continuum-transition regime. We also discuss the limitations of our model due to its various physical assumptions and we outline ideas for further development.