Eu's generalized hydrodynamics with its derived constitutive model: Comparison to Grad's method and linear stability analysis

Abstract

Recently, the generalized hydrodynamic equations (GHE) and nonlinear coupled constitutive relation (NCCR) model have been successfully utilized for the practical application in stable numerical computations of the non-equilibrium flows. However, their stability property has never been studied theoretically, and the inherent connection with classical Grad's moment still remains unclear. In order to clarify these issues, Eu's method, including the modeling of the non-equilibrium distribution function and the cumulant expansion for collision terms, is revisited to derive the modified moment system. A comparison of Eu's moment method with existing Grad's is presented in detail from the perspectives of distribution function and closure theory. The original infinite system of Eu's distribution function is first truncated into a finite system with 13 moments. Then through our attempt of Taylor expanding the truncated distribution function, a connection between Eu's distribution and Maxwellian and Grad's is established. Subsequently, a truncated closure method is conducted to clarify the relation between Eu's moment and Grad's moment equations. Finally, linear stability analysis of GHE and NCCR model is performed in one-dimensional and multi-dimensional processes, which shows that the equations are unconditionally stable for all wavenumbers and frequencies in the equilibrium rest state (ui0=0) and uniform-moving state (ui0≠0). The linear stability of GHE and NCCR model assures their numerical stability in the multi-dimensional computations, which can be deemed as one of the major benefits of Eu's theory.