Kinetic description of flow detachment at a micro-step

We study the pressure-driven steady gas flow over a backward-facing step in a two-dimensional microchannel. Focusing on the near-free-molecular regime of high Knudsen numbers, the problem is analyzed asymptotically based on the Bhatnager, Gross and Krook kinetic model, and supported by numerical Discrete Velocity Method computations. The wall conditions are formulated using the Maxwell model, superposing specular and diffuse surface conditions. The asymptotic solution contains the leading-order free-molecular description and a first-order integral representation of the near-free-molecular correction. In contrast with the common view that flow detachment is a low Knudsen number phenomenon, our results indicate that flow separation at the step may occur at arbitrarily large (yet finite) Knudsen numbers in smooth (specular reflecting) channels, where the flow is driven by temperature differences between the channel inlet and outlet reservoirs. It is then shown that detachment is significantly suppressed by the imposition of density variations between the reservoirs and an increase in the channel walls accommodation coefficient towards diffuse-surface conditions. While the mass flow rate in a specular channel decreases with decreasing Knudsen in a density-driven setup (in line with the Knudsen Paradox), it increases in a temperature-driven flow. The results are obtained for arbitrary differences between the inlet and outlet reservoir equilibrium properties, and are rationalized using the linearized problem formulation.