Simulation of interfacial instabilities during solidification-I. Conduction and capillarity effects

A combined Eulerian-Lagrangian numerical method is developed for simulating deformed interfaces arising in the solidification of pure materials. The interface tracking procedure employs marker particles and is the Lagrangian component of the calculation. The field equations are solved in a fixed Eulerian framework, so that the interface passes through the grid layout. Information from the explicitly tracked interface is used to apply boundary conditions at the exact interface location in each computational cell, in contrast with other Eulerian schemes. Consistent with the the established theory, in the absence of surface tension, the present simulations result in different types of behavior such as tip-splitting and cusp formation. For low surface tensions, due to the lack of physical length scales, the solutions are qualitatively affected by grid resolution with no unique solution available. In contrast, with substantial surface tension values the initial perturbation grows to form long fingers. The finger shapes reflect the stabilizing effects of capillarity. Unique solutions can be reached with nonzero surface tension.