Bulk viscosity of a binary mixture: the role of the intra-species interaction

The bulk viscosity $\zeta$ is a transport coefficient which is of central importance for various areas of modern physics. In particular, its determination for a mixture of more than one fluid is challenging, since it involves a complex interplay of multiple microscopic processes that operate on different time scales. Within the Chapman-Enskog framework, based on a series expansion of the Boltzmann distribution function, many previous works have derived the 1$^{\text{st}}$ order result for the $\zeta$ of a mixture. However, such a result fails to reproduce relevant physical features of the system, especially when the masses of the two components are similar. In this work we improve the 1$^{\text{st}}$ order Chapman-Enskog result by deriving the $\zeta$ at the 2$^{\text{nd}}$ order in the expansion. We show that this improved formula encodes many physical properties that the 1$^{\text{st}}$ order result misses: under specific conditions, the 2$^{\text{nd}}$ order result can be qualitatively and quantitatively very different from the 1$^{\text{st}}$ order one. Moreover, this result is compared against the $\zeta$ evaluated within the Green-Kubo formalism, by means of a numerical solution of the Relativistic Boltzmann equation. The agreement with respect to this benchmark is significantly improved when moving from the 1$^{\text{st}}$ to the 2$^{\text{nd}}$ order CE result.