Consistent, essentially conservative and balanced-force Phase-Field method to model incompressible two-phase flows

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In the present work, the Cahn-Hilliard Phase-Field model of incompressible two-phase flows is considered. Conditions needed for consistency of reduction, consistency of mass and momentum transport, and consistency of mass conservation are proposed. The mass flux in the Navier-Stokes equations is defined such that it satisfies the proposed consistency conditions. The analysis in both continuous and discrete levels shows that violation of the consistency conditions result in unphysical solutions and the inconsistent errors are proportional to the density contrast of the fluids. After considering the conservative form of the inertial term, a consistent and conservative scheme for momentum transport is developed. The balanced-force algorithm for the sharp interface model is extended to the surface force derived from the Cahn-Hilliard model. The proposed scheme is formally 2nd-order accurate in both time and space, satisfies the consistency conditions, conserves mass globally and momentum essentially, and is balanced-force, in the discrete level. Its convergence to the sharp interface solution is systematically discussed in cases including large density and viscosity ratios, surface tension, and gravity. Various two-phase flow problems with large density ratios are performed to validate and verify the proposed scheme and excellent agreements with published numerical and/or experimental results are achieved. The proposed scheme is a practical and accurate tool to study two-phase flows, especially for those including large density ratios.