Boundary-integral simulations of containerless solidification

We carry out boundary-integral simulations of a two-dimensional liquid droplet surrounded by air and solidified from a cool point on the boundary. There are three interfaces in the problem: solid-liquid, air-liquid, and air-solid. All three evolve in time in such a way that certain tri-junction conditions must be satisfied. Our numerical method describes the quasi-steady evolution of the interfaces in the limit of zero surface energy on the solidification front. A new iterative technique is developed to describe the interface evolution when mass and total energy are conserved and the local tri-junction conditions are satisfied at every instant in time. A method is also developed for efficient numerical integration over the interfaces by taking advantage of analytical formulas for Green's functions. We start the simulations by studying the case of equal densities of the solid and liquid. This allows us to verify the numerical method and obtain some estimates of the speed of the solidification front. Solid-liquid interface flattening is observed at the intermediate stages of solidification. When the densities of the two phases are different, elongated solidified particles are observed when the solid density is smaller than the liquid density. At the final stages of solidification, a corner is formed in agreement with observations in related experiments.