  In this work, a ternary phase-field model for two-phase flows in complex
geometries is proposed. In this model, one of the three components in the
classical ternary Cahn-Hilliard model is considered as the solid phase, and
only one Cahn-Hilliard equation with degenerate mobility needs to be solved due
to the condition of volume conservation, which is consistent with the standard
phase-field model with a single-scalar variable for two-phase flows. To depict
different wetting properties at the complex fluid-solid boundaries, the
spreading parameters in ternary phase-field model are determined based on the
Young's law, in which the liquid-solid surface tension coefficient is assumed
to be a linear function of gas-liquid surface tension coefficient and related
to the contact angle and the minimum curvature of the solid surface. In
addition, to achieve a high viscosity in the solid phase and preserve the
velocity boundary conditions on the solid surface, the phase-field variable of
the solid phase is also used to derive the modified Navier-Stokes equations. To
test the present model, we further develop a consistent and conservative
Hermite-moment based lattice Boltzmann method where an adjustable scale factor
is introduced to improve the numerical stability, and conduct the numerical
simulations of several benchmark problems. The results illustrate that present
model has the good capability in the study of the two-phase flows in complex
geometries.
