  In the present study, a consistent and conservative Phase-Field model is
developed to study thermo-gas-liquid-solid flows with liquid-solid phase
change. The proposed model is derived with the help of the consistency
conditions and exactly reduces to the consistent and conservative Phase-Field
method for incompressible two-phase flows, the fictitious domain Brinkman
penalization (FD/BP) method for fluid-structure interactions, and the
Phase-Field model of solidification of pure material. It honors the mass
conservation, defines the volume fractions of individual phases unambiguously,
and therefore captures the volume change due to phase change. The momentum is
conserved when the solid phase is absent, but it changes when the solid phase
appears due to the no-slip condition at the solid boundary. The proposed model
also conserves the energy, preserves the temperature equilibrium, and is
Galilean invariant. A novel continuous surface tension force to confine its
contribution at the gas-liquid interface and a drag force modified from the
Carman-Kozeny equation to reduce solid velocity to zero are proposed. The issue
of initiating phase change in the original Phase-Field model of solidification
is addressed by physically modifying the interpolation function. The
corresponding consistent scheme is developed to solve the model, and the
numerical results agree well with the analytical solutions and the existing
experimental and numerical data. Two challenging problems having a wide range
of material properties and complex dynamics are conducted to demonstrate the
capability of the proposed model.
