Lattice Boltzmann Method for Steady Gas Flows in Microchannels with Imposed Slip Wall Boundary Condition

In this paper, we use a lattice Boltzmann model (LBM) for simulation of rarefied gas flows in microchannels at high and moderate Knudsen numbers. The lattice Boltzmann method uses D2Q9 lattice structure and BGK collision operator with single relaxation time. The solid wall boundary conditions used in this paper are based on the idea of bounceback of the nonequilibrium part of particle distribution in the normal direction to the boundary. The same idea is implemented at inlet and exit boundaries as well as at the wall surfaces. The distribution functions at the solid nodes are modified according to imposed density and slip velocity values at the wall boundaries. Simulation results are presented for microscale Couette and Poiseuille flows. The results are validated against analytical and/or experimental data for the slip velocity, nonlinear pressure drop and mass flow rate at various flow conditions. It was observed that the current application of LBM can accurately recover the physics of microscale flow phenomena in microchannels. The type of boundary treatment used in this study enables the implementation of coupled simulations where the flow properties at the regions near the wall can be obtained by other numerical methods such as the Direct Simulation Monte Carlo method. ∗Address all correspondence to this author. INTRODUCTION With the advances in micromachining technology it has been possible to manufacture micron sized devices with high precision. When the characteristic length scale in a microsystem becomes comparable to the mean free path (λ) the flow becomes rarefied. The rarefaction of the flow is described using the nondimensional parameter called the Knudsen number, which is given as Kn = λ H where λ = κT Pσ2π √ 2 is the mean free path and H is the characteristic length. According to Bird [1], rarefied gas flows are classified into slip flow for 0.01 < Kn < 0.1, and transition flow for 0.1 < Kn < 10. For Kn < 0.01 the fluid can be considered as a continuum, whereas for Kn > 10 it is considered to be in free molecular regime. Recently, the lattice Boltzmann method (LBM) became a popular molecular-based modeling tool that can be used to simulate rarefied gas flow for the whole flow regime [2]. Nie et al. [3] applied the LBM method for microchannel flows and showed that LBM can capture behaviors such as velocity slip, nonlinear pressure distribution along the channel and dependence of mass flow rate on Knudsen number. Using a specular bounce back boundary condition, Lim et al. [4] showed that LBM results for the slip microchannel flow were in good agreement with experimental data for pressure distributions. It was also observed by Zhang et al. [5] that LBM can predict the correct trend of mass flow rate as the Knudsen number increases along the microchannel. 1 Copyright c © 2007 by ASME The purpose of this study is to examine the steady gas flow in microchannels using a lattice Boltzmann method with imposed wall boundary conditions. First the LBM method used in this work is summarized and modifications for rarefied flows is discussed. Next the numerical test cases is described and the boundary conditions are specified. The pressure distribution, velocity profiles and mass flow rates are presented in the results section. LATTICE-BOLTZMANN METHOD (LBM) The LBM model used in this study follows the evolution of density distribution function fa by calculating the lattice Boltzmann equation with streaming and single relaxation time BGK (Bhatnagar-Gross-Krook) collision operator [6] fa(x+ ea∆t, t +∆t) = fa(x, t)− ∆t τ ( fa(x, t)− f eq a (x, t)) . (1) In Eq. (1), x denotes the position vector and a stands for one of the 9 possible directions on the D2Q9 lattice structure shown in Fig. 1. The components of the equilibrium distribution function f eq a are given by f eq a (x) = waρ(x) [ 1+3 ea ·u c2 + 9 2 (ea ·u)2 c4 − 3 2 u2 c2 ]