  In this work, we consider a general consistent and conservative phase-field
model for the incompressible two-phase flows. In this model, not only the
Cahn-Hilliard or Allen-Cahn equation can be adopted, but also the mass and the
momentum fluxes in the Navier-Stokes equations are reformulated such that the
consistency of reduction, consistency of mass and momentum transport, and the
consistency of mass conservation are satisfied. We further develop a lattice
Boltzmann (LB) method and show that through the direct Taylor expansion, the
present LB method can correctly recover the consistent and conservative
phase-field model. Additionally, if the divergence of the extra momentum flux
is seen as a force term, the extra force in the present LB method would include
another term which has not been considered in the previous LB models. To
quantitatively evaluate the incompressibility and the consistency of the mass
conservation, two statistical variables are introduced in the study of the
deformation of a square droplet, and the results show that the present LB
method is more accurate. The layered Poiseuille flow and a droplet spreading on
an ideal wall are further investigated, and the numerical results are in good
agreement with the analytical solutions. Finally, the problems of the
Rayleigh-Taylor instability and dam break with the high Reynolds numbers and/or
large density ratios are studied, and it is found that the present consistent
and conservative LB method is robust for such complex two-phase flows.
