Vicente Garzó, J. Aaron Murray, Francisco Vega Reyes
The mass flux of a binary mixture of smooth inelastic hard spheres or disks is determined by solving the Boltzmann equation by means of the Chapman-Enskog method to first order in the spatial gradients. As in the elastic case, the associated transport coefficients $D$, $D_p$ and $D'$ are given in terms of the solutions of a set of coupled linear integral equations which are approximately solved by considering the first and second Sonine approximations. The diffusion coefficients are explicitly obtained as functions of the coefficients of restitution and the parameters of the mixture (masses, diameters and concentration) and their expressions hold for an arbitrary number of dimensions. In order to check the accuracy of the second Sonine correction for highly inelastic collisions, the Boltzmann equation is also numerically solved by means of the direct simulation Monte Carlo (DSMC) method to determine the mutual diffusion coefficient $D$ in some special situations (self-diffusion problem and tracer limit). As expected, the second Sonine approximation to $D$ improves the predictions made from the first Sonine approximation. Beyond the above two limits, the quantitative variation of the complete set of diffusion coefficients across the parameter space is demonstrated by comparing the first and second Sonine approximations. A discussion on the convergence of the Sonine polynomial expansion is also carried out. Finally, as an application of the results derived here, the theoretical expressions for the diffusion coefficients are used to analyze segregation induced by a thermal gradient. The results obtained in this paper extend previous works carried out in the tracer limit (vanishing mole fraction of one of the species) by some of the authors of the present paper.
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http://arxiv.org/abs/1210.2239
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