Discrete velocity modelling of gaseous mixture flows in MEMS

Abstract

The need of developing advanced micro-electro-mechanical systems (MEMS) has motivated the study of fluid-thermal flows in devices with micro-scale geometries. In many MEMS applications the Knudsen number varies in the range from 10 −2 to 10 2. This flow regime can be treated neither as a continuum nor as a free molecular flow. In order to describe these flows it is necessary to implement the Boltzmann equation (BE) or simplified kinetic model equations. The aim of the present work is to propose an efficient methodology for solving internal flows of binary gaseous mixtures in rectangular channels due to small pressure gradients over the whole range of the Knudsen number. The complicated collision integral term of the BE is substituted by the kinetic model proposed by McCormack for gaseous mixtures. The discrete velocity method is implemented to solve in an iterative manner the system of the kinetic equations. Even more the required computational effort is significantly reduced, by accelerating the convergence rate of the iteration scheme. This is achieved by formulating a set of moment equations, which are solved jointly with the transport equations. The velocity profiles and the flow rates of three different binary mixtures (He–Ar, Ne–Ar and He–Xe) in 2D micro-channels of various height to width ratios are calculated. The whole formulation becomes very efficient and can be implemented as an alternative methodology to the classical method of solving the Navier–Stokes equations with slip boundary conditions, which in any case is restricted by the hydrodynamic regime.