Kinetic description of flow detachment at a smooth micro-step: the near-free-molecular regime

Abstract

We study the pressure-driven steady gas flow, imposed by temperature or density gradients, over a backward-facing step in a two-dimensional microchannel. Focusing on the near-free-molecular regime of high Knudsen (\(\textrm{Kn}\)) numbers, the problem is analyzed asymptotically based on the Bhatnagar, Gross and Krook kinetic model, and supported by numerical Discrete Velocity Method and Direct Simulation Monte Carlo calculations. 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. Our results indicate that flow separation at the step can occur at arbitrarily large (yet finite) Knudsen numbers in channels with specular surfaces (i.e., having an accommodation coefficient of \(\alpha = 0\)), driven by temperature differences between the inlet and outlet reservoirs. It is then shown that detachment is significantly suppressed by density variations between reservoirs and partially diffuse surfaces (with \(\alpha \gtrsim 0.3\)). While the mass flow rate in a specular channel decreases with decreasing \(\mathrm {Kn\gg 1}\) 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.