  Varieties of energy-stable numerical methods have been developed for
incompressible two-phase flows based on the Navier-Stokes-Cahn-Hilliard (NSCH)
model in the Eulerian framework, while few investigations have been made in the
Lagrangian framework. Smoothed particle hydrodynamics (SPH) is a popular
mesh-free Lagrangian method for solving complex fluid flows. In this paper, we
present a pioneering study on the energy-stable SPH discretization of the NSCH
model for incompressible two-phase flows. We prove that this SPH method
inherits mass and momentum conservation and energy dissipation properties at
the fully discrete level. With the projection procedure to decouple the
momentum and continuity equations, the numerical scheme meets the
divergence-free condition. Some numerical experiments are carried out to show
the performance of the proposed energy-stable SPH method for solving the
two-phase NSCH model. The inheritance of mass and momentum conservation and the
energy dissipation properties are verified numerically.
