Simulation of Gaseous Microscale Transport Phenomena via Kinetic Theory

Kinetic theory of gases, as described by the Boltzmann or model kinetic equations, provides a solid theoretical approach for solving microscale transport phenomena in gases. Due to significant advancement in computational kinetic theory and due to the availability of high speed parallel computers, kinetic equations may be solved numerically with modest computational effort. In this framework, recently developed upgraded discrete velocity algorithms for solving linear and nonlinear kinetic equations are presented. In addition, their applicability in simulating efficiently and accurately multidimensional micro flow and heat transfer problems is demonstrated. Analysis and results are valid in the whole range of the Knudsen number. INTRODUCTION Gaseous microflows is a significant chapter in the emerging field of microfluidics [1, 2]. In general, these flows do not have local equilibrium and they are described by different length and time scales associated with different laws of physics. As it is demonstrated in the present work mesoscale approaches based on kinetic theory [3, 4] are capable of handling such problems in a unified manner. The parameter, which quantifies the departure of the gas from local or global equilibrium flow conditions, is the Knudsen number, which is defined as the ratio of the mean free path λ (i.e. the distance that particles travel between collisions) over a characteristic macroscopic length of the problem ( L / Kn L λ = ) [1]. When the Knudsen number is much less than one, the mean free path is so small that the gas may be considered as a continuum medium and the well-known Navier-Stokes equations can be applied to model the flow. This flow regime is known as the hydrodynamic regime and the solution of the governing equations can be obtained very efficiently implementing advanced numerical approaches. Although the validity of the Navier-Stokes equations breaks down for , it is possible to extend their applicability by substituting the no slip with suitable slip boundary conditions [1]. It has been found that in the so-called slip regime, defined by , the Navier-Stokes equations subject to the velocity slip and temperature jump boundary conditions may provide reliable results. Also, by introducing more advanced high-order slip boundary conditions the validity of the continuum equations may be extended to a wider range of the Knudsen number. However, it is important to note that the implementation of advanced reliable slip boundary conditions depends on the accurate estimation of the slip coefficients, which are obtained only through kinetic theory. Other attempts to facilitate and extend the implementation of macroscopic conservation equations are based on the application of more complicated constitutive laws yielding a set of generalized hydrodynamic equations [1, 5]. It is obvious however, that all this effort is limited by the hydrodynamic assumption and can not be valid in the whole range of the Knudsen number. However, it is fully justified by the fact that there is a lot of knowledge and experience on the numerical solution of nonlinear hydrodynamic equations and whenever applicable the gain in computational effort is significant. 3 10 Kn − > 3 10 0.1 Kn − ≤ ≤ At the other end, when the Knudsen number is much greater than one, and more specifically for , the mean free path is so large that collisions between molecules and boundaries occur more often than collisions between molecules. This flow regime is known as the free molecular regime and in this case it may be considered that each particle travels independently of each other, ignoring the intermolecular collisions. Due to this simplification it is possible following the particle paths and based on the method of characteristics to yield closed form solutions for simple flow configurations. In more complex geometry the Test Particle Monte Carlo (TPMC) method is applied with great success [6]. 10 Kn ≥ Finally, when the Knudsen number has intermediate values ( 0.1 10 Kn < < ) the gas may not be considered as a continuum medium, neither as a medium consisting of individual particles. 1 Copyright © 2010 by ASME This flow regime is known as the transition regime and it may be modeled by kinetic theory as it is described by the Boltzmann equation or alternatively by simplified kinetic model equations, where the primary unknown is the particle distribution function [3, 4, 7]. Then, the macroscopic quantities of practical interest are easily obtained by taking moments of the distribution function. This is a mesoscale approach since it is characterized by phase space volumes and times, which are small compared to the spatial and time scales on which macroscopic quantities vary but large enough to contain a sufficient number of molecules to allow a statistical description. A kinetic description is self-contained providing that the laws of intermolecular interaction are specified. Traditionally, the computational solution of the kinetic equations is much more demanding and complicated than the solution of the hydrodynamic equations. It is important to note that in the transition and free molecular regimes new transport phenomena, known as cross effects, arise and then they diminish gradually in the slip and hydrodynamic regimes. In gaseous micro devices the Knudsen number may rise due to small length scales or due to low pressures, while all flow regimes may occur in the same micro system. A typical way to circumvent, when needed, the numerical solution of the kinetic equations, is the implementation of the Direct Simulation Monte Carlo method [6]. The DSMC method is a statistical computational approach where the region of the gas flow is divided into a large number of cells having dimensions such that the change in flow properties across each cell is small. Then, the evolution in space and time of a large number of randomly selected and statistically representative molecules in each cell is considered. The computational molecules, each of which represents a huge number of real molecules move, interact with solid boundaries and collide to each other following basic kinetic principals, so as to statistically mimic the behavior of real molecules. The DSMC method due to its simplicity has attracted considerable attention and although, in general, it requires large computer memory and long CPU time, is by far the most widely used approach when the flow is in the whole range of the Knudsen number. However, in low speed microflows as well as in high frequency unsteady microflows, both of which are quite common in microfluidics, despite the significant improvements and upgrades which have been achieved [8], the DSMC method requires considerably increased computation effort due to statistical noise. Therefore, it is always reasonable to search for reliable alternatives capable of solving any microflow configuration in an accurate and computationally efficient manner. Such an alternative methodology may be the fully deterministic solution of suitable kinetic model equations [9]. This approach is well developed and advanced in the field of rarefied gas dynamics and can be applied in a straight forward manned in microflows due to the fact that a gas microflow may be considered as a rarefied gas flow. Of course, fully deterministic numerical solutions of kinetic equations, consisting of the discretization of the distribution function in the physical and molecular velocity spaces, are complicated. However nowadays, due to the availability of high speed parallel computers and due to the significant advancement in computational kinetic theory made during the last years, kinetic equations may be solved numerically in an efficient manner. In the case of multi dimensional configurations deterministic numerical schemes include variational, integro-moment and discrete velocity methods [10]. Over the years the discrete velocity method (DVM) [11, 12, 13, 14] and its recently introduced accelerated version [15, 16] have shown to be very efficient numerical algorithms providing accurate results with modest computational effort. They have been applied in a series of linear problems including pressure, temperature and concentration driven fully developed flows of single gases and gaseous mixtures through long channels of various cross sections. Most of this work is based on the BGK [17], Shakhov [10] and McCormack [18, 19] kinetic models, although the numerical solution of the Boltzmann equation in several occasions has been achieved [20, 21, 22]. It may be stated that in the case of low Mach number microflows the implementation of linearized kinetic theory is the most computationally efficient approach. Fully deterministic algorithms based on the discrete velocity method have been also applied in the case of nonlinear configurations including the cylindrical Couette flow [23, 24], flow in a cavity due to discontinuous wall temperature [25] and half space evaporation and condensation [26]. Very recently a methodology has been proposed to upgrade the computational efficiency of the nonlinear DVM solvers [27] in order to be more competitive with respect to the DSMC method even at high speed flows. Within this framework, in this paper, recently developed upgraded discrete velocity algorithms for solving linear and nonlinear kinetic equations are presented. In addition, their applicability in simulating efficiently and accurately micro transport phenomena is demonstrated by reviewing recent results of rarefied gas flows through channels and heat transfer through rarefied gases confined between cylinders. Analysis and results are valid in the whole range of the Knudsen number. In Section 2 the fundamental kinetic equations with the associated boundary conditions are provided. In Section 3 the basic and the accelerated discrete velocity algorithms are described. Results for the specific problems under consideration with the relative discussion are given in Section 4, followed by brief concluding remarks in Section 5. GOVERNING EQUATIONS At the kinetic (or mesoscale) level the state of a monoatomic gas is described by the distribution function ( ) , , f t r ξ , defined such that ( ) , , f t d d r ξ r ξ is the number of molecules which, at time have positions lying within the volume about and velocities lying within a velocityelement about . The evolution of the distribution function , t