# The effect of Knudsen layers on rarefied cylindrical Couette gas flows

## Abstract

We investigate a power-law probability distribution function to describe the mean free path of rarefied gas molecules in non-planar geometries. A new curvature-dependent model is derived by taking into account the boundary-limiting effects on the molecular mean free path for surfaces with both convex and concave curvatures. The Navier–Stokes constitutive relations and the velocity-slip boundary conditions are then modified based on this power-law scaling through the transport property expressions in terms of the mean free path. Velocity profiles for isothermal cylindrical Couette flow are obtained using this power-law model and compared with direct simulation Monte Carlo (DSMC) data. We demonstrate that our model is more accurate than the classical slip solution, and we are able to capture important non-linear trends associated with the non-equilibrium physics of the Knudsen layer. In addition, we establish a new criterion for the critical accommodation coefficient that leads to the non-intuitive phenomenon of velocity inversion. The power-law model predicts that the critical accommodation coefficient is significantly lower than that calculated using the classical slip solution, and is in good agreement with available DSMC data. Our proposed constitutive scaling for non-planar surfaces is based on simple physical arguments and can be readily implemented in conventional fluid dynamics codes for arbitrary geometric configurations of microfluidic systems.

## Keywords

Molecular mean free path Knudsen layer Cylindrical Couette flow Velocity inversion Curvature effects Gas micro flows Rarefied gas dynamics## Notes

### Acknowledgments

The research leading to these results has received funding from the European Community’s Seventh Framework Programme FP7/2007-2013 under grant agreement ITN GASMEMS no. 215504. Additional support was provided by the UK Engineering and Physical Sciences Research Council (EPSRC) under the auspices of Collaborative Computational Project 12 (CCP12).

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