Microscale gas flow : a comparison of Grad's 13 moment equations and other continuum approaches




Thatcher, Toby

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Advances in manufacturing techniques over the last decade have made it possible to make electrical devices with dimensions as small as 90 nanometers [I]. Using similar techniques, devices that perform moving mechanical tasks less than 100 pm are being manufactured in quantity [2] [3], e.g., pumps, turbines, valves and nozzles. These devices are incorporated into microelectromechanical systems (MEMS) that can be potentially used in devices such as medical and chemical sensors, and fuel cells. The gas and fluid flows in devices of this size exhibit behavior that can not be described by the classical Navier-Stokes and Fourier equations of continuum mechanics. This happens when flows become rarefied such that the mean free path (distance between two subsequent particle collisions) is not negligible compared to the characteristic length scale. The rarefaction of a fluid flow is also seen in the upper atmosphere for larger length scales, e.g., for re-entry for space craft and some supersonic jet aircraft. Currently, when one looks to model fluid flow and heat transfer in a rarefied flow there are two predominantly accepted choices. Either one uses jump and slip boundary conditions with the Navier-Stokes and Fourier (NSF) equations, or a statistical particle model such as direct simulation Monte-Carlo (DSMC) [4] and the Boltzmann equation. DSMC is computationally intensive for complex flows and the NSF solutions are only valid for low degrees of rarefaction. As an alternative to these methods we have used Grad's 13 moment expansion of the Boltzmann equation [5]. For its implementation, a set of boundary conditions and three numerical methods for the solution have been devised. The model is applied to the solution of 2-D micro Couette flow with heat transfer. Results are compared to those obtained from the Navier-Stokes-Fourier equations, reduced Burnett equations, Regularized 13 moment equations and DSMC simulations.