Transition point prediction in a multicomponent lattice Boltzmann model: Forcing scheme dependencies

Pseudopotential-based lattice Boltzmann models are widely used for numerical simulations of multiphase flows. In the special case of multicomponent systems, the overall dynamics are characterized by the conservation equations for mass and momentum as well as an additional advection diffusion equation for each component. In the present study, we investigate how the latter is affected by the forcing scheme, i.e., by the way the underlying interparticle forces are incorporated into the lattice Boltzmann equation. By comparing two model formulations for pure multicomponent systems, namely the standard model [X. Shan and G. D. Doolen, J. Stat. Phys. 81, 379 (1995)] and the explicit forcing model [M. L. Porter et al., Phys. Rev. E 86, 036701 (2012)], we reveal that the diffusion characteristics drastically change. We derive a generalized, potential function-dependent expression for the transition point from the miscible to the immiscible regime and demonstrate that it is shifted between the models. The theoretical predictions for both the transition point and the mutual diffusion coefficient are validated in simulations of static droplets and decaying sinusoidal concentration waves, respectively. To show the universality of our analysis, two common and one new potential function are investigated. As the shift in the diffusion characteristics directly affects the interfacial properties, we additionally show that phenomena related to the interfacial tension such as the modeling of contact angles are influenced as well.

Figure 1

Density profiles for two immiscible fluids. As the interface is smeared over several lattice nodes, its position can be obtained with ${\rho }_1-{\rho }_2=0$ [50].

Figure 2

Diffusion coefficients for both STD and EF model for different relaxation times $\tau =1,0.9,0.8$. The simulation results (points) are in excellent agreement with the theoretical predictions (solid lines) from Eq. (15).

Figure 3

Dissolved and bulk densities of component 1 (unfilled) as well as the corresponding, measured interfacial tensions (filled) for different potential functions for the STD and EF model in (a) and (b), respectively. The transition point, above which two stable phases with dissimilar densities develop, is in excellent agreement with the theoretically predicted values (indicated by vertical lines). The critical values are $G_{\mathrm{crit},{\mathrm{\Psi }}_{\rho }}=0.5$, $G_{\mathrm{crit},{\mathrm{\Psi }}_{\exp }}\approx 2.15$, and $G_{\mathrm{crit},{\mathrm{\Psi }}_{\mathrm{atan}}}=\pi$ for the STD model and twice as large for the EF model. Below $G_{\mathrm{crit}}$, there are no interfaces and the interfacial tension is zero (neglected). In the immiscible regime, $\sigma$ scales differently with $G$ for different potential functions.

Figure 4

Relation of $G$ and the interfacial tension $\sigma$ using $\mathrm{\Psi }={\mathrm{\Psi }}_{\rho }$ and $\tau =1.0,1.5,2.0$. While the interfacial tension depends on the relaxation time for the STD model, it does not for the EF model (in accordance with Ref. [34]).

Figure 5

Dissolved and bulk densities of component 1 for different relaxation times for the STD model with $\mathrm{\Psi }={\mathrm{\Psi }}_{\rho }$. As predicted by Eq. (14), the transition point scales with $\tau$. The critical values are $G_{\mathrm{crit},\tau =1}=0.5$, $G_{\mathrm{crit},\tau =0.9}=4/9$, and $G_{\mathrm{crit},\tau =0.8}=0.375$.

Figure 6

Contact angles for both model variations with ${\mathrm{\Psi }}_{\rho }$ and ${\mathrm{\Psi }}_{\exp }$ in (a) and (b), respectively. In all cases, we set $\tau =1.0$ and $G_{\mathrm{red}}=1.8$. As a result of the shifted diffusion characteristics, the values of $G_2^{\mathrm{wall}}$ must be significantly larger to achieve the same contact angle.