Extended gas dynamic equations with multiple translational temperature

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)] 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 [CD. Levermore, J. Stat. Phys 83, pp. 1021 (1996)]. 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. Our gas dynamic equations are natural extensions of the Navier-Stokes equations to the near continuum flow regime and can be used for microflow computations.