  In this work, we proposed a diffuse interface model for the dendritic growth
with thermosolutal convection. In this model, the sharp boundary between the
fluid and solid dendrite is replaced by a thin but nonzero thickness diffuse
interface, which is described by the order parameter governed by the
phase-field equation for the dendritic growth. The governing equations for
solute and heat transfer are modified such that the previous special treatments
for source term can be avoided. To solve the model for the dendritic growth
with thermosolutal convection, we also developed a diffuse-interface
multi-relaxation-time lattice Boltzmann (LB) method. In this method, the order
parameter in the phase-field equation is combined into the force caused by the
fluid-solid interaction, and the treatment on the complex fluid-solid interface
can be avoided. In addition, four LB models are developed for the phase-field
equation, concentration equation, temperature equation and the Navier-Stokes
equations in a unified framework. Finally, to test the present
diffuse-interface LB method, we performed some simulations of the dendritic
growth, and found that the numerical results are in good agreements with some
previous works.
