  In this work, a phase-field model is developed for the dendritic growth with
gas bubbles in the solidification of binary alloys. In this model, a total free
energy for the complex gas-liquid-dendrite system is proposed through
considering the interactions of gas bubbles, liquid melt and solid dendrites,
and it can reduce to the energy for gas-liquid flows in the region far from the
solid phase, while degenerate to the energy for thermosolutal dendritic growth
when the gas bubble disappears. The governing equations are usually obtained by
minimizing the total free energy, but here some modifications are made to
improve the capacity of the conservative phase-field equation for gas bubbles
and convection-diffusion equation for solute transfer. Additionally, through
the asymptotic analysis of the thin-interface limit, the present general
phase-field model for alloy solidification can match the corresponding free
boundary problem, and it is identical to the commonly used models under a
specific choice of model parameters. Furthermore, to describe the fluid flow,
the incompressible Navier-Stokes equations are adopted in the entire domain
including gas, liquid, and solid regions, where the fluid-structure interaction
is considered by a simple diffuse-interface method. To test the present
phase-field model, the lattice Boltzmann method is used to study several
problems of gas-liquid flows, dendritic growth as well as the solidification in
presence of gas bubbles, and a good performance of the present model for such
complex problems is observed.
