Analysis of force treatment in the pseudopotential lattice Boltzmann equation method

In this paper, different force treatments are analyzed in detail for a pseudopotential lattice Boltzmann equation (LBE), and the contribution of third-order error terms to pressure tensor with a force scheme is analyzed by a higher-order Chapman-Enskog expansion technique. From the theoretical analysis, the performance of the original force treatment of Shan-Chen (SC), Ladd, Guo et al., and the exact difference method (EDM) are ɛ_{Ladd}<ɛ_{Guo}<ɛ_{EDM}≤ɛ_{SC} with the relaxation time τ≥1, while ɛ_{Ladd}<ɛ_{Guo}<ɛ_{SC}<ɛ_{EDM} with τ<1; here ɛ is a parameter related to the mechanical stability and the subscripts are the corresponding force scheme. To be consistent with the thermodynamic theory, a force term is introduced to modify the coefficients in the pressure tensor. Some numerical simulations are conducted to show that the predictions of modified force treatment of the pseudopotential LBE are all in good agreement with the analytical solution and other predictions.