Extended Gas Dynamic Equations for Microflows
Kun Xu and Zhaoli Guo
Abstract: In an early approach, we proposed a kinetic model with multiple translational temperature [K. Xu, H. Liu and J. Jiang, Phys. Fluids 19, 016101 (2007) (doi:10.1063/1.2429037)] to simulate non-equilibrium flows. In this paper, instead of using three temperatures in the x-, y-, and z-directions, we further define the translational temperature as a second-order symmetric tensor. Based on a multiple stage BGK-type collision model and the Chapman–Enskog expansion, the corresponding macroscopic gas dynamics equations in three-dimensional space will be derived. The zeroth-order expansion gives the 10 moment closure equations of Levermore [C.D. Levermore, J. Stat. Phys 83, pp.1021–1065 (1996) (doi:10.1007/BF02179552)]. The derived gas dynamic equations can be considered as a regularization of Levermore's 10 moments equations in the first-order expansion. Our new gas dynamic equations have the same structure as the Navier–Stokes equations, but the stress-strain relationship in the Navier–Stokes equations is replaced by an algebraic equation with temperature differences. At the same time, the heat flux, which is absent in Levermore's 10 moment closure, is recovered. As a result, both the viscous and the heat conduction terms are unified under a single anisotropic temperature concept. In the continuum flow regime, the new gas dynamic equations automatically recover the standard Navier–Stokes equations. The extended gas dynamic equations present a closed system, which is a natural extension of the Navier-Stokes system to the near continuum flow regime and can be used for microflow computations. Both analytical and numerical solutions for the force-driven Poiseuille flow and the Couette flow in the transition flow regime will be used to validate the current system.