On the modelling of isothermal gas flows at the microscale

Lockerby, Duncan A. and Reese, Jason M. (2008) On the modelling of isothermal gas flows at the microscale. Journal of Fluid Mechanics, 604. pp. 235-261. ISSN 0022-1120

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    This paper makes two new propositions regarding the modelling of rarefied (non-equilibrium), isothermal gas flows at the micro scale. The first is a new test case for benchmarking high-order, or extended, hydrodynamic models for these flows. This standing, time-varying shear wave problem does not require boundary conditions to be specified at a solid surface, so is useful for assessing whether fluid models can capture rarefaction effects in the bulk flow. We assess a number of different proposed extended hydrodynamic models, and we find the R13 equations perform the best in this case. Our second proposition is a simple technique for introducing non-equilibrium effects caused by the presence of solid surfaces into the computational fluid dynamics framework. By combining a new model for slip boundary conditions with a near-wall scaling of the Navier-Stokes constitutive relations, we obtain a model that is much more accurate at higher Knudsen numbers than the conventional second-order slip model. We show this provides good results for combined Couette/Poiseuille flow, and that the model can predict the stress/strain-rate inversion that is evident from molecular simulations. The model's generality to non-planar geometries is demonstrated by examining low-speed flow around a micro-sphere. It shows a marked improvement over conventional predictions of the drag on the sphere, although there are some questions regarding its stability at the highest Knudsen numbers.