This work investigates the Cahn–Hilliard equation for interface capturing in two-phase flows, focusing on the widely used summation constraint requiring all order parameters to sum to unity. We develop and compare two formulations: a one-equation approach that solves for a single component while inferring the second via the constraint, and a two-equation approach that solves for both components independently. A free energy functional for the two-equation formulation is proposed, recovering the classical single-component form when the constraint is applied. Linear stability analysis and benchmark simulations using the lattice Boltzmann method reveal that the two-equation scheme exhibits superior numerical stability by eliminating the asymmetry introduced by the summation constraint. However, both schemes yield nearly identical results in the quasi-static limit where velocity effects are negligible. Computationally, the one-equation scheme runs ∼25% faster and requires ~11% less memory, highlighting a fundamental trade-off between efficiency and numerical robustness in phase-field modeling of binary fluids.
Haghani et al. (Sun,) studied this question.