To obtain a better understanding of the evolution of macrosegregation during the direct chill casting of aluminum alloys, a binary mixture model has been developed that accounts for the redistribution of alloying elements through the transport of free-floating dendrites and fluid flow in the melt and mushy zones of a solidifying ingot. Separate and distinct mixture momentum equations are employed to account for momentum transfer in two-phase regions which include a slurry of free-floating dendrites and a rigid solid matrix. A scaling analysis revealed that the flow of liquid and free-floating dendrites is controlled by a balance between buoyancy forces arising from mixture density variations and the development of an adverse pressure gradient. In a companion paper, the model is used to predict macrosegregation in Al-4.5 wt% Cu and Al-6.0 wt% Mg billets.