Finite-difference-based multiple-relaxation-times lattice Boltzmann model for binary mixtures

In this paper, we propose a finite-difference-based lattice Boltzmann equation (LBE) model with multiple-relaxation times (MRT), in which the distribution functions of individual species evolve on a same regular lattice without any interpolations. Furthermore, the use of the MRT enables the model more flexible so that it can be applied to mixtures of species with different viscosities and adjustable Schmidt number. Some numerical tests are conducted to validate the model, the numerical results are found to agree well with analytical solutions/or other numerical results, and good numerical stability of the proposed LBE model is also observed.

Figure 1

Shear viscosity as a function of the relaxation rate $s_{a8}$ with $s_{b8}^{-1}=0.55$, $X_a=0.7$ and $X_b=0.3$. Symbols are the FD-MRT results, and solid lines are the theoretical predictions given by Eq. (47).

Figure 2

Shear viscosity as a function of the relaxation rate $s_{b8}$ with $s_{a8}^{-1}=0.55$, $X_a=0.7$ and $X_b=0.3$. Symbols are the FD-MRT results, and the solid lines are the theoretical predictions given by Eq. (47).

Figure 3

Self-diffusivity as a function of the relaxation rate. Symbols are the FD-MRT results, and solid lines are theoretical predictions given by Eq. (57).

Figure 4

Density profiles of the species at different times. Symbols are the FD-MRT results, and solid lines are the predictions of Ref. [16] at $t=0$, 0.02, 0.05, and 0.1 ms, respectively.

Figure 5

Concentration profiles of the species at different times. Symbols are the LBE results, and solid lines are theoretical predictions given by Eq. (68) at time step $10000$ (○);$40000$ (▽), $80000$ (◻), $160000$ (△).

Figure 6

Transport coefficients of the measured $\nu$ and $D_{ab}$. Asterisk means the numerical error in Schmidt number is lower than 5%.