The pseudo-potential lattice Boltzmann (LB) model is a widely used multiphase model in the LB community. In this model, an interaction force, which is usually implemented via a forcing scheme, is employed to mimic the molecular interactions that cause phase segregation. The forcing scheme is therefore expected to play an important role in the pseudo-potential LB model. In this paper, we aim to address some key issues about forcing schemes in the pseudo-potential LB model. Firstly, theoretical and numerical analyses will be made for Shan-Chen's forcing scheme and the exact-difference-method (EDM) forcing scheme. The nature of these two schemes and their recovered macroscopic equations will be shown. Secondly, through a theoretical analysis, we will reveal the physics behind the phenomenon that different forcing schemes exhibit different performances in the pseudo-potential LB model. Moreover, based on the analysis, we will present an improved forcing scheme and numerically demonstrate that the improved scheme can be treated as an alternative approach for achieving thermodynamic consistency in the pseudo-potential LB model.

We perform Langevin simulations on the depinning dynamics of two-dimensional magnetized colloids on a random substrate. On increasing the magnetic field strength, we find for the first time a crossover from plastic to smectic flows as well as a crossover from smectic to elastic crystal flows above depinning. For both the smectic and elastic crystal flows, a power-law scaling relationship could be obtained between the average velocity and applied driving force. The scaling exponent is found to be larger than 1 for smectic flow. But, for the elastic crystal flow, the scaling exponent is found to be less than 1. For the plastic flow, no power-law scaling relationship between the average velocity and applied driving force can be derived and history dependence of the depinning occurs. Within the crossover from plastic to smectic flows, a sudden decrease in the critical driving force is observed, and a sudden increase is found in the critical driving force across the crossover from smectic to elastic crystal flows, accompanied by a crossing of the curves of average velocity versus driving force.