Lattice Boltzmann modeling of microchannel flow in slip flow regime

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Abstract

We present the lattice Boltzmann equation (LBE) with multiple relaxation times (MRT) to simulate pressure-driven gaseous flow in a long microchannel. We obtain analytic solutions of the MRT-LBE with various boundary conditions for the incompressible Poiseuille flow with its walls aligned with a lattice axis. The analytical solutions are used to realize the Dirichlet boundary conditions in the LBE. We use the first-order slip boundary conditions at the walls and consistent pressure boundary conditions at both ends of the long microchannel. We validate 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. As expected, the LBE results agree very well with IP-DSMC and DSMC results in the slip velocity regime, but deviate 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. Possible extensions of the LBE for transition flows are discussed.

Introduction

The rapidly growing interest in micro-electro-mechanical systems (MEMS) has also stimulated a great interest in modeling and simulation methods for microflows [1], [2]. In particular, the gaseous flow through a long microchannel has become an important test case for various numerical methods, because of its important applications in microdevices [2]. The physics of this flow has been well studied by now [2], [3], [4]. Even at very low Mach number, gaseous flows are often compressible in microdevices because of substantial pressure drops and density variations caused by viscous effects. In a long, constant-area microchannel, the degree of rarefaction and the Mach number increase along the channel, thus all Knudsen number regimes may be encountered, and the pressure drop becomes nonlinear. In essence, compressibility and rarefaction are the key characteristics of gaseous flows through a long microchannel [3].

Among various methods, the lattice Boltzmann equation (LBE) has been advocated as an effective means for microflows simulations because of its kinetic origin [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. Based on a theoretical analysis and numerical evidence, we justify the application of the LBE method for simulation of microflows in the slip flow regime. We will focus our effort on pressure-driven flows through a long microchannel.

Numerous LBE studies have been devoted to pressure-driven microchannel flows [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. Several interesting observations can be made. First of all, the lattice Bhatnagar-Gross-Krook (LBGK) model [22], [21] is exclusively used in these works, although it is well known that the boundary conditions in the LBGK models depend on the viscosity [23], [24] and that the so-called “slip velocity” observed in the LBGK models with bounce-back boundary conditions is a numerical artifact [25]. Second, regardless of the value of the Knudsen number Kn, the profile of the streamwise velocity u is a linear superposition of a perfect parabola and a constant slip velocity Us at boundaries, and the magnitude of Us depends on Kn. This is inconsistent with the solutions of the full or linearized Boltzmann equations. It is thus no surprise that the LBGK model yields qualitatively incorrect results [5] for microchannel flow at the Knudsen number Kn=0.194 [26]. Because numerical analysis and convergence studies have not been shown in the LBE studies [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], nor have sufficient validation and verification been carried out, the validity of the LBE microflow modeling is in question [26].

In this work we intend to demonstrate the capability of the LBE method for microflow simulations. Through both theoretical analysis and numerical simulation, we will show that the LBE method can simulate microchannel flow. In particular, we obtain the analytic solutions of the LBE with various fluid–solid boundary conditions for the incompressible Poiseuille flow with its walls parallel to a lattice axis. Our analysis elucidates that the erroneous results in the previous LBE simulations are mostly due to the deficiencies in the LBGK model coupled with the bounce-back boundary conditions [25], [24]. We will demonstrate that the LBE models with multiple relaxation times (MRT) [27], [28], [29] can overcome certain inconsistencies in the LBGK models with bounce-back boundary conditions. Although the present study only deals with the simple case of the two-dimensional Poiseuille flow with its walls aligned with a lattice axis, the boundary conditions for geometries with arbitrary curvatures have been proposed [31], [23], [24], [30].

The remaining part of this paper includes a summary of known results for gaseous microchannel flow in Section 2; a brief exposition of the LBE method in Section 3; the analytic solutions of the LBE for the Poiseuille flow with various boundary conditions in Section 4; a description of boundary conditions used in the LBE simulations in Section 5; the numerical results for microchannel flow, compared with the slip Navier–Stokes solution, the DSMC and IP-DSMC data up to Kn=0.388 in Section 6; and a concluding discussion in Section 7.

Section snippets

Theory of pressure-driven flow through a long microchannel

The two-dimensional isothermal flow through microchannels with low Mach number can be analyzed by using the Navier–Stokes equations [3], [2]. A succinct summary of the known results relevant to the present work is provided below. For a long channel with a height H and length LH, the flow variables can be represented as perturbation series of the aspect ratioε:=H/L1.The dimensionless parameters in the flow are the Reynolds number Re, the Mach number Ma and the Knudsen number Kn, all defined by

Lattice Boltzmann equation with multiple relaxation times

The lattice Boltzmann equation (LBE) is a simple explicit algorithm, which can be derived from the linearized Boltzmann equation, and is often associated with a square or cubic lattice, rjZD, on which the discretized single particle distribution function fi(rj,tn):=f(rj,ci,tn) evolves. The particle velocity space ξ is discretized into a symmetric discrete velocity set {ci}=-{ci} so that in discrete time tn:=δtN0=δt{0,1,2,}, fictitious particles represented by {fi} move synchronously from one

Analytic solution of LBE for the incompressible Poiseuille flow

It is instructive to show the analytic solutions of the LBE given by Eq. (7) for the incompressible Poiseuille flow in 2D. The setup is the following. The channel height is H in the spanwise direction y with Ny grid points, i.e., j=1, 2, , Ny. A constant body force is applied in the x direction on all fluid nodes. Since we use periodic boundary conditions in the streamwise direction, the channel length L is an irrelevant parameter for the time being. Two horizontal solid walls are placed

Boundary conditions in LBE simulations

Correct boundary conditions are crucially important in the LBE simulations of microchannel flows. Two types of boundary conditions must be dealt with: (a) the boundary conditions at the walls and (b) pressure boundary conditions at the inlet and the outlet. In our simulations, the system size is (Nx+2)×(Ny+2), i.e., i=0, 1, , Nx, Nx+1, and j=0, 1, , Ny, Ny+1. The fluid nodes are 1iNx and 1jNy. The extra nodes beyond the fluid region are used to store data before advection.

We use the

Numerical results

In what follows, we will demonstrate that the lattice Boltzmann equation with the MRT collision model can indeed simulate microchannel flow with finite Knudsen number effects. The boundary conditions described in the previous section are used in the simulations. The values of the relaxation rates are: si=sν=1/τ for i5 and 7, and s5=s7=sq, where sν is chosen to keep μ constant and sq is related to sν by Eq. (19).

Conclusions

In this work we implement the athermal MRT-LBE with a first-order slip velocity model and demonstrate that the LBE method can be used to simulate gaseous flow through a long microchannel in the slip flow regime. The LBE results agree well with the slip Navier–Stokes, DSMC, and IP-DSMC results in the slip flow regime. While the LBE and slip Navier–Stokes results agree well with each other in the transition-flow regime, they deviate significantly from the DSMC results, as expected.

We would like

Acknowledgments

The authors are indebted to Dr. Z.L. Guo for providing the DSMC and IP-DSMC data of Shen et al. [26] and helpful discussions. LSL is grateful to Dr. I. Ginzburg for many constructive discussions and suggestions. FV would like to acknowledge the support from the Research Foundation-Flanders (FWO-Vlaanderen). LSL would like to acknowledge the support from the National Science Foundation of the US through the Grants CBET-0500213 and DMS-0807983, and NASA Langley Research Center through a C&I grant

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