Modelling and Simulation in Engineering

Volume 2016, Article ID 7619746, 9 pages

http://dx.doi.org/10.1155/2016/7619746

## Extended Macroscopic Study of Dilute Gas Flow within a Microcavity

Département de physique, Université Moulay Ismaïl, Meknès, Morocco

Received 27 August 2016; Accepted 24 October 2016

Academic Editor: Ricardo Perera

Copyright © 2016 Mohamed Hssikou et al. This 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.

#### Abstract

The behaviour of monatomic and dilute gas is studied in the slip and early transition regimes using the extended macroscopic theory. The gas is confined within a two-dimensional microcavity where the longitudinal sides are in the opposite motion with constant velocity . The microcavity walls are kept at the uniform and reference temperature . Thus, the gas flow 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. The macroscopic gas proprieties are studied for different values of the so-called Knudsen number (Kn), which gives the gas-rarefaction degree. The 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 Systems (MEMS) has greatly developed and they have wide areas of application [1–3]. This fast growth of MEMS use is not followed enough by the physical understanding of rarefied gas flows 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 often defy the predictions made using the scaling laws developed for large systems. In fact, the gas flows inside the MEMS, under the standard conditions, are usually characterized by a mean free path comparable to the system characteristic-length . Thus, the so-called Knudsen number of gas flow is in the slip-transition regimes range; that is, . In this case, the conventional computational fluid dynamics (CFD) scheme, based on the classical Navier-Stokes and Fourier (NSF) equations, becomes inappropriate to describe the gas flow behaviour in MEMS devices. Therefore, the Knudsen number, in MEMS, is not sufficiently 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 effects can be found in the problems of gas flows under low pressure and atmospheric conditions [5]. For gas flows outside the hydrodynamic regime ( [6], many interesting rarefaction effects such as velocity-slip and temperature-jump at the walls [7–10], Knudsen paradox, Knudsen layers [11], transpiration flow [12, 13], thermal stress [14], and heat flux 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. The direct simulation Monte Carlo (DSMC) is the largely kinetic method used to simulate a rarefied gas flow where the behaviour is mainly described by the Boltzmann equation [16]. The accuracy of this method is proved by many previous studies especially with the actual computers capabilities. But, the computational cost and fluctuations noises, especially in the low-signal flows, 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 first 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 finite wave speeds, and discontinuous subshock structures when the Mach number lies above [18]. We also note that the Grad 13-moment equations, for nonlinear problems, lack suitable boundary conditions. But, when the rarefaction degree becomes more intense, the Knudsen number value is in the range of , and a purely kinetic approach is needed for describing the gas flows [19].

Keeping the benefits of both approaches and to avoid their failures, Struchtrup has adopted recently the combination of the above approaches [19]. This leads to the set of regularized 13-moments equations (R13) used as higher order of continuum solution, that is, , to capture the rarefaction effects, described by the nonlinear terms. The main goal of this paper is to investigate the behaviour of a dilute gas flow inducing only the longitudinal shear stress using the classical theory of NSF, with slip and jump boundary conditions, and the regularized 13-moment equations of Grad approaches. In this study, the rarefaction effects are evaluated in the slip and early transition regimes range. We assume that the gas flow is induced with no synergetic contributions from external force fields.

#### 2. Statement of Problem

A monatomic and Maxwell-molecules gas, where the collisions rate is independent of the collision-patterns velocities, is confined within a square microcavity. In this gas, the particles are interacting via a potential with being the interparticles distance [16]. The orthogonal cross section and the origin of the coordinate system are shown in Figure 1. The upper and lower sides are in the opposite motion with a constant velocity . The microcavity walls are kept at uniform and environmental temperature . The macroscopic proprieties of the gas are evaluated for different values of the Knudsen number in the slip and early transition regime, .