BGK models for diffusion in isothermal binary fluid systems

Two Bhatnagar–Gross–Krook (BGK) models for isothermal binary fluid systems—the classical single relaxation time model and a split collision term model—are discussed in detail, with emphasis on the diffusion process in perfectly miscible ideal gases. Fluid equations, as well as the constitutive equation for diffusion, are derived from the Boltzmann equation using the method of moments and the values of the transport coefficients (viscosity and diffusivity) are calculated. The Schmidt number is found to be equal to one for both models. The split collision term model allows the two fluid components to have different values of the viscosity, while the single relaxation time model does not have this characteristic. The value of the viscosity does not depend on the density in the split collision term model, as expected from the classical kinetic theory developed by Maxwell. Possible extension of BGK models to non-ideal gases and ideal solutions (where the Schmidt number is larger than 1) is also investigated.