Abstract
We evaluate the accuracy of the McCormack model by comparing its solutions for Couette and Fourier flows of binary gaseous mixtures with results from the linearized Boltzmann equation. Numerical simulations of Ne–Ar and He–Xe gas mixtures are carried out from slip to near free-molecular flow regimes for different values of the molar concentration. Our numerical results show that while there are only small differences in the shear stress in Couette flow and the heat flux in Fourier flow, calculated from the two kinetic equations, differences in other macroscopic quantities can be very large, especially in free-molecular flow regime. Moreover, the difference between results from the two models increases with the molecular mass ratio and the molar concentration of the heavier species. Finally, the applicability of the McCormack model, which was derived for linearized flows only, is investigated by comparing its solutions with those from the Boltzmann equation for Fourier flow with large wall-temperature ratios.
Highlights
In practical applications like vacuum technology, porous media, and the chemical industry, information about the heat/mass transfer in rarefied gaseous mixtures is indispensable
(HS) molecules has been solved by an analytical version of the discrete-ordinates (ADO) method [8], and the accuracy of the McCormack model has been assessed for a He–Ar mixture: the McCormack model produces accurate shear stress for each species, and the velocity of the heavier species [5,8]; the velocity of the lighter species and especially the heat flux significantly deviate from the linearized Boltzmann equation (LBE) results
Our numerical results showed that when only the shear stress in Couette flow and the heat flux in Fourier flow are required, the McCormack model can be used, as the differences in the results from the two kinetic models are within 2%
Summary
In practical applications like vacuum technology, porous media, and the chemical industry, information about the heat/mass transfer in rarefied gaseous mixtures is indispensable. Couette and Fourier flows between two parallel plates are chosen as benchmark test cases as they are classical problems in fluid mechanics. Solutions for single-species gases have been well studied, few papers have investigated gaseous mixtures. The plane Couette flow of a binary gaseous mixture was first studied [1,2,3,4,5] using kinetic models for the Boltzmann equation (BE), such as the Hamel model [6] for Maxwellian molecules and the McCormack model for general intermolecular potentials [7]. Following the McCormack model, the influence of intermolecular interactions on the velocity and shear stress in three mixtures (Ne–Ar, He–Ar, and He–Xe) [4] and the influence of the gas-surface interaction on the flow properties were investigated [5].
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