A note on the summation relation in phase-field equations

In this paper, we investigate phase-field interface capturing equations for two-fluid systems to probe their accuracy and computational cost. Two different schemes are considered: In the first scheme, one of the two order parameters is numerically solved based on a phase-field equation, while the other order parameter is determined through the summation relation; the summation of order parameters equals unity. In the second scheme, the two order parameters are both obtained numerically by solving their respective phase-field equations. A phase-field model based on the color-gradient (CG) method is chosen, and available lattice Boltzmann models are employed for solving the interface-capturing equations together with the hydrodynamic equation. It is shown that for the first scheme, which includes the summation relation, numerical results become asymmetrical. Also, in some cases, it results in nonphysical interfaces. In terms of computational resources, this first scheme is about 11% faster with 25% less computational memory usage than the second scheme. It is shown that only for a zero velocity domain do the two schemes lead to equal results. Also, a theoretical analysis is conducted to highlight the differences between the two approaches.