Research Article Extended Macroscopic Study of Dilute Gas Flow within a Microcavity Mohamed Hssikou, Jamal Baliti, and Mohammed Alaoui D´ epartement de physique, Universit´ e Moulay Isma¨ ıl, Mekn` es, Morocco Correspondence should be addressed to Mohamed Hssikou; hssikoumed@gmail.com Received 27 August 2016; Accepted 24 October 2016 Academic Editor: Ricardo Perera Copyright © 2016 Mohamed Hssikou et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te behaviour of monatomic and dilute gas is studied in the slip and early transition regimes using the extended macroscopic theory. Te gas is confned within a two-dimensional microcavity where the longitudinal sides are in the opposite motion with constant velocity ±  . Te microcavity walls are kept at the uniform and reference temperature  0 . Tus, the gas fow is transported only by the shear stress induced by the motion of upper and lower walls. From the macroscopic point of view, the regularized 13-moment equations of Grad, R13, are solved numerically. Te macroscopic gas proprieties are studied for diferent values of the so-called Knudsen number (Kn), which gives the gas-rarefaction degree. Te results are compared with those obtained using the classical continuum theory of Navier-Stokes and Fourier (NSF). 1. Introduction Recently, the technology of the Microelectromechanical Sys- tems (MEMS) has greatly developed and they have wide areas of application [1–3]. Tis fast growth of MEMS use is not followed enough by the physical understanding of rarefed gas fows in these microdevices. For this purpose, several studies have been recently focused on for more understanding of the physical phenomena involved in these small devices [4]. In fact, the performances of MEMS ofen defy the predictions made using the scaling laws developed for large systems. In fact, the gas fows inside the MEMS, under the standard conditions, are usually characterized by a mean free path  comparable to the system characteristic-length . Tus, the so-called Knudsen number Kn ∼ / of gas fow is in the slip-transition regimes range; that is, 0.001 < Kn ≤10. In this case, the conventional computational fuid dynamics (CFD) scheme, based on the classical Navier-Stokes and Fourier (NSF) equations, becomes inappropriate to describe the gas fow behaviour in MEMS devices. Terefore, the Knudsen number, in MEMS, is not sufciently small to guarantee the validity of the NSF equations and the processes in MEMS need to be modelled with more accurate transport models. Similar rarefaction efects can be found in the problems of gas fows under low pressure and atmospheric conditions [5]. For gas fows outside the hydrodynamic regime (Kn > 0.001) [6], many interesting rarefaction efects such as velocity-slip and temperature-jump at the walls [7–10], Knudsen paradox, Knudsen layers [11], transpiration fow [12, 13], thermal stress [14], and heat fux without temperature gradients can take place [15]. Hence, there is a pressing need to develop the more accurate methods allowing a good description of gas- dynamic processes into these microsystems. Te direct sim- ulation Monte Carlo (DSMC) is the largely kinetic method used to simulate a rarefed gas fow where the behaviour is mainly described by the Boltzmann equation [16]. Te accuracy of this method is proved by many previous studies especially with the actual computers capabilities. But, the computational cost and fuctuations noises, especially in the low-signal fows, remain the major inconveniences of this kinetic method [17]. Indeed, many macroscopic approaches are proposed such as the Chapman-Enskog (CE) expansion and the Grad moments theory. At the frst order of CE both approaches lead to the famous laws of Navier-Stokes and Fourier. However, on one hand, the instability of Burnett equations obtained at second order of CE expansion is the main problem of this approach. On the other hand, the Grad 13-moment equations are hyperbolic in nature, yielding fnite Hindawi Publishing Corporation Modelling and Simulation in Engineering Volume 2016, Article ID 7619746, 9 pages http://dx.doi.org/10.1155/2016/7619746