document: The particular phase field model we employ is an extension of [6], and is based on the three dimensional thermal phase field model of [7] and two dimensional thermal-solutal phase field model of [8]. One feature of the physical problem is that it is purely dissipative, or entropy increasing, as all natural relaxational phenomena are. The resulting PDEs are of Allen–Cahn [9] and Carn–Hilliard type [10]. That is to say, the model involves time derivatives of the three fields coupled to forms involving variational derivatives of some functional – typically the free energy functional. As the dendrite grows the free energy reduces monotonically with time but never achieves equilibrium if the domain boundary is far from the dendrite. Although we have listed some of the difficult aspects of this model, the relaxational aspect is typically an asset and results in stable numerical schemes: there is no convection, for example (at least in the absence of flow in the melt).
extractive_keyphrases: dendrite free energy functional functional melt numerical schemes pdes phase field model time derivatives of the three fields coupled to forms involving variational derivatives
abstractive_keyphrases: three dimensional thermal phase field model two dimensional thermal-solutal phase field model