# Micro-electro-mechanical systems

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### INTRODUCTION

### Background

Micro-channels and micro-orifice are among the most important integral parts which require some attention and study for good performance of many micro-electro-mechanical systems (MEMS). Micro-electro-mechanical systems (MEMS) are devices with a characteristic length of less than 1 mm but more than 1 µm; and hence for the air flows the flow Knudsen number (Kn) in such devices is characterized between 0.01 and 10, which is in the range of slip flow and transitional flow regime. The micro flows can be divided into different flow regimes based on the Kn which is a dimensionless number defined as the ratio of the length of molecular mean free to a representative physical length where the flow is defined scale can be seen in Table I.

For Kn < 0.01 the flow falls under continuum flow regime that is governed by continuum hypothesis; as Kn increases the domination of continuum hypothesis for the flow starts to vanish. For 0.01 < Kn < 0.1 the flow is called slip flow regime and at this flow regime, there exists a slip velocity at the wall boundary. For 0.1 < Kn < 10 the flow becomes transitional regime. These two flow regimes of slip and transitional are typical for gas flows in micro flows. Beyond Kn = 10, the flow is considered as free molecular flow regime that should be considered by a particle based analysis that solves the equations for motions of molecules. A number of numerical analyses have been performed for investigating micro flows through micro-channels and micro-filters. The correct slip-velocity boundary condition implementation is an important task in simulating the slip flow regime. Because of rarefaction effects, the no-slip boundary condition cannot be implemented on solid boundaries. For these conditions the flows through micro-tubes and micro-channels have been investigated experimentally and analytically. To achieve reliable solutions, numerical and experimental studies require more accurate methods and techniques, especially in micro-channels. Analytical study can be considered an alternative approach to alleviate the preceding difficulties. In micro gas flow, there are some important effects which cannot be predicted in the normal viscous flow based on the continuum hypothesis Karniadakis, et al. [1]. As micro flows have dimensions of the order of 0.1 to 10 µm, at this very small scale; for Kn in the range of 0.1 - 10 a fluid particle can travel a relatively long distance and collide with a boundary before it collides with another particle. As Kn increases this rarefaction effect becomes truly significant. Another main effect is compressibility, which results in a pressure distribution along the flow direction which is not linear. Gas flows are fundamentally a compressible flow and there is a need for a large pressure difference to drive a fluid through a relatively long channel compared with macro-scale channels consequently this compressibility become important under the large pressure variation between the inlet and outlet Agrawal and Agrawal [2]. Viscous heating and thermal creep Karniadakis, et al. [1] can be other important effects but the current work does not cover these effects. Ahmed and Beskok [3]have investigated viscous heating as well as compressibility and rarefaction effects in micro gas flow through micro-filters. An important effect in micro gas flows is the slip velocity at the boundary wall. The present work is mainly focused on obtaining this slip velocity at the wall by applying proper boundary conditions to gain appropriate results for micro channels and micro filters. Lee and Lin [4] regarded the slip velocity at the boundary wall from the results of LBM simulation as a numerical error produced by the lack of stability of applied boundary conditions. However this slip velocity is obviously an existing phenomenon in the real physics of micro flows. Gaseous slip flow was verified analytically by Karniadakis, et al. [1], who presented a unified flow model for micro flows which is applicable under the wide Knudsen number range including the transitional flow regime. Lee, et al. [4] applied the boundary conditions to generate slip velocity in a physically proper way. Besides, other researchers (Zhang, et al. [5],Tang, et al. [6],Lim, et al. [7]) conducted numerical simulation using various types of boundary conditions to analyze the slip velocity at the wall under various conditions. The Lattice Boltzmann method is a computational scheme which can be used to model the different flow regimes which are classified by the Kn. It is a modification of Lattice Gas Automata and can represent the fluid dynamics at the microscopic and kinetic regimes. The LB method deals with the populations of particles statistically rather than using the Eulerian or Lagrangian approaches which are generally used for fluid description. The LB method can be implemented using the finite differencing schemes which can have the particle streaming step followed by collision of the particles. The LB approach to fluid flow mainly deals with the concept of mass and momentum conservation. The transport coefficients for LB methods depend on the time step and lattice spacing which can be looked at as lattice viscosity and numerical viscosity. In the LB method the particle populations are forced to move with the velocities which are discrete vectors for the specific direction. A collision operator called the BGK collision operator which is inversely proportional to density of the fluid is used to redistribute the particle population. The LB method is discussed in detail later. Kuo and Chen [8] proposed a unified model to impose no-slip and slip boundary conditions which introduces the tangential momentum accommodation coefficient determining the change of tangential momentum on the wall which gives the results similar to that of Maxwell's first order slip model--------. Sofonea and Sekerka [9] studied flows in Couette and Poiseuille flow in micro-channels using bounce-back and diffuse reflection boundary conditions and verified the existence of slip which depends on the Kn. In the case of Poiseuille flow the slip velocity is found to depend upon the lattice spacing and Kn. Hecht and Harting [10] proposed an on-site boundary conditions that specify the exact position of the boundary, independent of other simulation parameters where the boundary condition acts locally, is independent of the details of the relaxation process during collision and contains no artificial slip i.e. does not contain any numerical slip. Ahmed and Hecht [11] extended Hecht, et al. [10] model and proposed a boundary condition with adjustable slip length and found that the slip length is independent of the shear rate and density but is proportional to the BGK relaxation time. Chen and Tian [12] studied the Langmuir slip model,---------- instead of the popularly used Maxwell slip model which is incorporated into the Lattice Boltzmann (LB) method through the non-equilibrium extrapolation scheme to simulate the rarefied gas flow. This model gave better results compared to that of the Maxwell slip model. Verhaeghe, et al. [13] studied the Lattice Boltzmann Equation (LBE) with Multiple Relaxation Times (MRT) to simulate pressure-driven gaseous flow in a long micro-channel. Analytical solutions of the MRT-LBE with various boundary conditions for the incompressible Poiseuille flow with its walls aligned with a lattice axis were studied. The analytical solutions were used to realize the Dirichlet boundary conditions---------- in the LBE. They used the first-order slip boundary conditions at the walls and consistent pressure boundary conditions at both ends of the long micro-channel. They validated the LBE results using the compressible Navier-Stokes (NS) equations with a first-order slip velocity, the information-preservation direct simulation Monte Carlo (IP-DSMC) and DSMC methods. Their LBE results agree very well with IP-DSMC and DSMC results in the slip velocity regime, but deviated significantly from IP-DSMC and DSMC results in the transition-flow regime in part due to the inadequacy of the slip velocity model, while still agreeing very well with the slip NS results----------. They proposed possible extensions of the LBE for transition flows Li and Kwok [14] proposed a lattice Boltzmann model in the presence of external force fields to describe micro-fluidic phenomena considering pressure as the only external force to drive liquid flow. Their results were in good agreement with recent experimental data in a pressure-driven micro-channel flow that could not be fully described by electro-kinetic theory. Their differences between the predicted and the experimental Reynolds numbers from pressure gradients were within 5%.

### Problem statement and presentation

Zea and Chambers [15] performed 2D LBM simulation for micro-channel and micro-orifice flows. The micro-orifice was the basic configuration of the micro-filter. Their simulations yielded compressibility and density distributions along the channel and filter that were matched well with the literature, but did not simulate the slip velocity at the wall of the micro-channel accurately. For the current LBM simulation of micro-channels, 212100 lattice nodes are used to make the ratio of length to height 100. Implementations are primarily performed in the slip flow regime to examine the slip velocity at the wall. To provide the slip velocity at the wall of the micro-channel and micro-filter, the no-slip bounce back, reflection factor () and accommodation coefficients () are applied to the boundary conditions. For the micro-orifice simulations, the ratio which is the ratio of the open orifice area to the total area was selected as 0.6 to compare the results with the literature. The shape of the orifice was selected as a square and cylindrical. Micro-orifice simulations are also conducted in the slip flow regime with the proper boundary conditions obtained for the micro-channel simulations. The air was assumed to flow through the micro-orifice under isothermal conditions throughout the computations. From the results obtained from the simulations the flow field is determined and this flow field is used to calculate the particle trajectories.

### Objectives of research.

The objective of this project is to investigate important micro-gas flow features including slip velocity, compressibility, and rarefaction and diffusion effects for micro-channel and micro-orifice flows using the LBM simulation.

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