Time consistency of lattice Boltzmann equation with an external force

When an external force is present, the particle distribution as well as the local velocity and momentum are continuously accelerated from the view of the Boltzmann equation. The continuous acceleration, even within a time step, causes the conventional lattice Boltzmann equation for nonideal fluids to lack the time consistency, leading to its evolution being inconsistent with the macroscopic system and its theoretical analysis being contradictive to the numerical simulation. This paper presents a time-consistent lattice Boltzmann equation that uses the mean effects to depict the continuous acceleration and the integral characteristic. It abides Boltzmann's assumption of the independence between particle collisions and external forces and resolves the contradiction between theoretical analyses and the numercial simulations. The time-consistent equation can recover the exact Navier-Stokes equations by the Chapman-Enskog analysis. It can simulate the liquid-gas coexistence densities and saturation properties, which are in perfect agreement with the theoretical predictions of the Maxwell equal-area law and the Kelvin equation, respectively. Finally, an improved scheme enables the time-consistent multiphase model to adjust the surface tension independently by the surface tension coefficient.