Enskog Theory for Polydisperse Granular Mixtures. III. Comparison of dense and dilute transport coefficients and equations of state for a binary mixture

TL;DR

This paper compares dense and dilute hydrodynamic theories for binary granular mixtures, showing significant differences in transport coefficients and equations of state, and provides corrected GHD expressions for better accuracy in dense regimes.

Contribution

It presents a detailed comparison of dense and dilute theories for granular mixtures, including corrected GHD equations and analysis of their applicability.

Findings

Dense theory predictions can differ by over 100% from dilute predictions.

The study provides corrected, self-contained GHD equations for dense granular mixtures.

Differences between theories increase with volume fraction and other parameters.

Abstract

The objective of this study is to assess the impact of a dense-phase treatment on the hydrodynamic description of granular, binary mixtures relative to a previous dilute-phase treatment. Two theories were considered for this purpose. The first, proposed by Garz\'o and Dufty (GD) [Phys. Fluids {\bf 14}, 146 (2002)], is based on the Boltzmann equation which does not incorporate finite-volume effects, thereby limiting its use to dilute flows. The second, proposed by Garz\'o, Hrenya and Dufty (GHD) [Phys. Rev. E {\bf 76}, 31303 and 031304 (2007)], is derived from the Enskog equation which does account for finite-volume effects; accordingly this theory can be applied to moderately dense systems as well. To demonstrate the significance of the dense-phase treatment relative to its dilute counterpart, the ratio of dense (GHD) to dilute (GD) predictions of all relevant transport coefficients and equations of state are plotted over a range of physical parameters (volume fraction, coefficients of restitution, material density ratio, diameter ratio, and mixture composition). These plots reveal the deviation between the two treatments, which can become quite large ($>$100%) even at moderate values of the physical parameters. Such information will be useful when choosing which theory is most applicable to a given situation, since the dilute theory offers relative simplicity and the dense theory offers improved accuracy. It is also important to note that several corrections to original GHD expressions are presented here in the form of a complete, self-contained set of relevant equations.