Fig. 1. Mesh distribution and velocity profiles of the Poiseuille flow with F = 0.001. Solid line, analytical solution; dot, numerical result.

Fig. 2. Numerical error of the Poiseuille flow at different F as a function of the number of grid in y direction.

Fig. 3. Sketch of backward-facing step, boundary conditions, and standing vortices.

Fig. 4. Variation in reattachment and separation lengths with inlet Reynolds number for single-phase flow.

Fig. 5. Variation in reattachment and separation lengths with Stokes number for Re = 450,700; α^inlet = 3 × 10⁻³.

Fig. 6. Variation in reattachment and separation lengths with void fraction for Stk = 1 × 10⁻³,1 × 10⁻²;Re = 450.

Fig. 7. Concentration field of void fraction for Re = 450, Stk = 1 × 10⁻³ and α^inlet = 3 × 10⁻³.

Fig. 8. Concentration field of void fraction for Re = 450, Stk = 1 × 10⁻² and α^inlet = 3 × 10⁻³.

Fig. 9. Variation in x₁, x₂ and x₃ with Re = 389, α^inlet = 3 × 10⁻³ and Stk = 3 × 10⁻³.

Fig. 10. Concentration plot of particles for Re = 389, α^inlet = 3 × 10⁻³ and Stk = 3 × 10⁻³ at t = 119.7.

Fig. 11. Concentration plot of particles for Re = 389, α^inlet = 3 × 10⁻³ and Stk = 3 × 10⁻³ at t = 1266.7.

Fig. 12. Time difference between the coupled LBM and the present FDLBE with various Re; Stk = 1 × 10⁻³, α^inlet = 3 × 10⁻³.

Fig. 13. The relative error differences between the coupled LBM and the present FDLBE with various Re; Stk = 1 × 10⁻³, α^inlet = 3 × 10⁻³.

Comparison of the critical Stokes number between the coupled LBM and the present model, α^inlet = 3 × 10⁻³