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A novel diffuse interface model is presented for the direct numerical simulation of microstructure evolution in solidification processes involving convection in the liquid phase. The solidification front is treated as a moving interface in the diffuse approximation as known from phase-field theories. The no-slip condition between the melt and the solid is realized via a drag resistivity in the diffuse interface region. The model is shown to accurately reproduce the usual sharp interface conditions in the limit of a thin diffuse interface region. A first test of the model is provided for flow through regular arrays of cylinders with a stationary interface. Then, two examples are presented that involve solid/liquid phase-change: (i) coarsening of a mush of a binary alloy, where both the interface curvature and the flow permeability evolve with time, and (ii) dendritic growth in the presence of melt convection with particular emphasis on the …