The phase-field model of solidification is extended to include the effects of fluid flow in the melt. The phase-field model is based on coupling the nist-equations for heat flow in the liquid and solid phases with an auxiliary nist-equation that describes the evolution of the phase-field variable, which is a non-conserved order parameter indicating the local phase, solid or liquid, at each point of the material. The solid-liquid interface is then represented by a diffuse transition layer in which the phase-field variable changes rapidly between its values in the bulk phases. The model is extended to include fluid flow by a further coupling to the Navier-Stokes nist-equations. Preliminary studies have been performed for a model in which the solid phase is treated as a liquid of high viscosity compared to the liquid phase. The main coupling in the Navier-Stokes nist-equations is then through an additional term in the stress tensor that depends on the gradients of the phase-field variable, representing the effects of capillary forces within the diffuse interface
