Appendix B: Hard Sphere Correction to Interaction Term

The usual lattice Boltzmann method assumes the fluid is composed of point particles. To include a volume exclusion interaction and in effect obtain a relative volume of the fluid particles we utilize an Enskog hard sphere model. The relative volume fraction can be determined from the sphere radius and the number density. The application of Enskog theory to multicomponent fluid mixtures is described by López de Haro et al. [118] and in references cited in this work. While there are different formulations of hard sphere models, such as standard Enskog theory (SET) and revised Enskog theory (RET), we will utilize a form of forcing, arising from hard sphere interactions that are treated to lowest order in density. In this case, the two theories are identical. Further details are described in reference [118]. In the isothermal regime, the additional correction to the forcing due to hard sphere collisions, B^i(HS) is

$$B^{i(HS)}_a =-\frac{2b_{ij} \chi_{ijc}}{n^j} n^{i(eq)}_a {\bf (e_a-v)} \cdot \frac{\partial n^j}{\partial \bf {x}}$$ (37)

where $\chi_{ijc}$ is the equilibrium value of the pair correlation function for spheres of species i and j at contact with the equilibrium density replaced by the total local equilibrium density at the point ${\bf x}$, $b_{ij}=\frac{2}{3}\pi n_j \sigma^3_{ij}/\rho$ with, $\sigma_{ij}$ equal to the distance between sphere centers in contact and $\rho$ is the local density. The total forcing on a fluid component i is then Bⁱ_a+B^i(HS)_a where Bⁱ_a is defined in Eq. 13.