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n an early approach, we proposed a kinetic model with multiple translationaltemperature [K. Xu, H. Liu and J. Jiang, Phys. Fluids {\bf 19}, 016101 (2007)],to simulate non-equilibrium flows. In this paper, instead of using threetemperatures in $x-$, $y-$, and $z$-directions, we are going to further definethe translational temperature as a second-order symmetric tensor. Based on amultiple stage BGK-type collision model and the Chapman-Enskog expansion, thecorresponding macroscopic gas dynamics equations in three-dimensional spacewill be derived. The zeroth-order expansion gives the 10 moment closureequations of Levermore [C.D. Levermore, J. Stat. Phys {\bf 83}, pp.1021(1996)]. To the 1st-order expansion, the derived gas dynamic equations can beconsidered as a regularization of Levermore's 10 moments equations. The new gasdynamic equations have the same structure as the Navier-Stokes equations, butthe stress strain relationship in the Navier-Stokes equations is replaced by analgebraic equation with temperature differences. At the same time, the heatflux, which is absent in Levermore's 10 moment closure, is recovered. As aresult, both the viscous and the heat conduction terms are unified under asingle anisotropic temperature concept. In the continuum flow regime, the newgas dynamic equations automatically recover the standard Navier-Stokesequations. The current gas dynamic equations are natural extension of theNavier-Stokes equations to the near continuum flow regime and can be used formicroflow computations. Two examples, the force-driven Poiseuille flow and theCouette flow in the transition flow regime, are used to validate the model.Both analytical and numerical results are encouraging.

Generalized Gas Dynamic Equations for Microflows