COMPARISON BETWEEN VS MODEL AND IPL AND LJ POTENTIALS Molecular Diffusion in a Homogeneous Heat-Bath

The variable sphere (VS) molecular model is introduced to Monte Carlo simulation of rarefied gas flow. The VS model provides the consistency for diffusion and viscosity cross-sections with those of any realistic intermolecular potential. The VS model has a much simpler scattering law than either the variable hard sphere (VHS) or variable soft sphere (VSS) models; also, it has the same computational efficiency as the VHS and VSS models. In this study, VS model is applied to the inverse power law (IPL) and Lennard-Jones (LJ) potentials. Comparative simulations between the VS mode, IPL and LJ potentials are made for the molecular diffusion in a homogeneous heat-bath gas, normal shock wave structure in a monatomic gas, and normal shock wave structure in binary gas mixture. The good agreement is shown in the molecular diffusion and shock wave structure between the VS model and the IPL and LJ potentials. The VS model is applied to the statistical inelastic cross-section (SICS) model for the rotational inelastic collision of the diatomic molecule. Comparative simulation between the VS model with the SICS (VS-SICS) model and VSS model with SICS (VSS-SICS) model is made for rotational relaxation in a nitrogen normal shock wave. The agreement in the shock wave structure and rotational energy distribution function across the shock wave between the VS-SICS model and the VSSSICS model are good. It is shown that the macroscopic transport phenomena are affected by the diffusion and viscosity cross-sections rather than the scattering law of the each molecular collision. INTRODUCTION The Monte Carlo method has been developed and widely applied to a variety of rarefied gas flow problems because of its accuracy, efficiency, and simplicity. For engineering, realistic and effective molecular collision models are required. The variable hard sphere (VHS) molecular model is attractive as a simple and efficient realistic collision model and applied to a lot of rarefied gas flow problems. The VHS model is suggested for the inverse power law (IPL) potential for the first time, and then extended to a more realistic intermolecular potential such as Lennard-Jones (LJ) potential. Previous papers, pointed out, however, that either the diffusion crosssection (diffusion coefficient) or the viscosity cross section (viscosity coefficient) of the VHS model is inconsistent with that of the potential, such as IPL and LJ potentials. In order to overcome this problem, the variable soft sphere (VSS) molecular model, is introduced for both the diffusion cross-section (diffusion coefficient) and viscosity cross-section (viscosity coefficient) to be consistent with any realistic intermolecular potential and applied to the IPL and LJ potentials. From the basic concepts of the VHS and VSS models, many simple and realistic molecular collision models should be constructed. In this study, the variable sphere (VS) molecular model is introduced for both diffusion and viscosity cross-sections to be consistent with any realistic intermolecular potential and applied to IPL and JL potentials. The scattering law of the VS model is simpler than that of the VHS and VSS models and computational efficiency is almost identical to VHS and VSS models. Comparative calculations between the VS model and IPL and LJ potentials are made for the molecular diffusion in a homogeneous heat-bath gas and normal shock-wave structure in a monatomic gas and normal shock wave structure in binary gas mixture. For the VS model for IPL potential, the Maxwell molecule and hard sphere are chosen as practical models for softness and hardness. The VS model for IPL potential is applied to the statistical inelastic cross-section (SICS) model and comparative calculation between the VS model with the SICS model (VS-SICS) and VSS model with SICS model (VSS-SICS) is made for rotational relaxation in a nitrogen normal shock wave structure. VARIABLE SPHERE MOLECULAR MODEL Scattering Law and Cross-Sections The VS molecular model has definite diameter d, which varies with molecular relative collision energy ε and scatters with the same deflection angle χ = χ0 for effective impact parameter b, defined as 0 ( ) 0 ( ) b d b d χ χ ≤  =  >  . (1) The total cross-section of the VS model is obtained as 2 2 t bdb d σ π π = = ∫ . (2) Diameter d (or total cross-section σt) and deflection angle χ0 are defined so that both diffusion and viscosity cross-sections are consistent with those of a realistic intermolecular potential. The VS model diffusion and viscosity cross-sections are defined as (1) 2 0 2 (1 cos ) (1 cos ) Q bdb d π χ χ π = − = − ∫ and (3) (2) 2 2 2 0 2 (1 cos ) (1 cos ) Q bdb d π χ χ π = − = − ∫ , (4) respectively. From equations (3) and (4), total cross-section and cosine of deflection angle χ0 are uniquely obtained as (1) 2 (1) (1) (2) 2 t Q d Q Q Q σ π = = − and (5)