Second-order cooling-rate coefficients for granular binary mixtures

Determine the complete set of second-order spatial-gradient coefficients in the cooling rate for granular binary mixtures, including the coefficients multiplying linear Laplacian terms, quadratic density and temperature gradients, and quadratic velocity-gradient terms, beginning with the low-density case.

Background

At second order in spatial gradients, the cooling rate contains coefficients associated with Laplacians of the species densities and temperature, products of density and temperature gradients, and quadratic velocity-gradient terms. The paper explains that evaluating these coefficients for mixtures is technically difficult and that neglecting them is an approximation used in the presented Navier–Stokes stability analysis. The unresolved status is particularly notable because explicit results were unavailable even for dilute binary mixtures.

References

The evaluation of the above set of coefficients for granular mixtures is a quite intricate problem. To the best of our knowledge, no explicit results for these coefficients have been reported for granular binary mixtures, even in the simplest case of a low-density mixture ($\phi=0$).

— Enskog kinetic theory in a model of confined granular mixtures. Heat flux and some applications  (2609.16868 - Méndez et al., 15 Sep 2026) in Section 4.2, paragraph following Eq. (4.13)