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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 operating point of the tip.