Abstract
We derive a multi-species BGK model with velocity-dependent collision frequency for a non-reactive, multi-component gas mixture. The model is derived by minimizing a weighted entropy under the constraint that the number of particles of each species, total momentum, and total energy are conserved. We prove that this minimization problem admits a unique solution for very general collision frequencies. Moreover, we prove that the model satisfies an H-Theorem and characterize the form of equilibrium.
Highlights
In this paper, we present a BGK-type model for gas mixtures that, in the case of two species, takes the form∂t f1 + v · ∇x f1 = ν11(M11 − f1) + ν12(M12 − f1), (1)∂t f2 + v · ∇x f2 = ν22(M22 − f2) + ν21(M21 − f2), along with appropriate boundary and initial conditions
There are many BGK models for gas mixtures proposed in the literature [1,5,10,12,14,15,16,23,28], many of which satisfy these basic requirements and, in addition, are able to match some prescribed relaxation rates and/or transport coefficients that come from more complicated physics models or from experiment
In the case of neutral gases, velocity independent collision frequency leads to transport properties in the fluid regime that are inconsistent with the full kinetic collision operator, e.g., the Prandtl number
Summary
We present a BGK-type model for gas mixtures that, in the case of two species, takes the form. In the single-species case, the original BGK model [2] serves this purpose It has the same collision invariants as the Boltzmann operator (which lead to conservation of number, momentum, and energy) and it satisfies an HTheorem. In the case of neutral gases, velocity independent collision frequency leads to transport properties in the fluid regime that are inconsistent with the full kinetic collision operator, e.g., the Prandtl number Models such as the ES-BGK model and the Shakov model make changes to the target Maxwellian to provide extra degrees of freedom to the system and correct this defect, but still retain the constant collision frequency assumption. Multispecies extension of these methods [18] penalize each species with a single relaxation operator rather than penalizing individual reaction pairs, which can lead to inaccuracies in cases where some collisional combinations are more important than others Another concern is the computational cost of a collision operator evaluation.
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