The unidirectional solidification of a dilute alloy (tin-bismuth) in a two-dimensional rectangular cavity is investigated. A uniform computational grid is employed. A vorticity-vector potential representation is used for the governing equations for the velocity field. The interface compatibility conditions for vorticity and vector potential are developed and discussed. The energy equation is solved for the temperature field, while the species equation is solved for the solute distribution. The constitutive equations are solved using a true transient method. The solution scheme incorporates an Alternating Direction Implicit (ADI) approximation for the vorticity while the conjugate gradient method is employed for the vector potential equation. The results obtained from the numerical simulations compare very well with experimental results for directional solidification in the literature in terms of the propagation of the solidification front as well as the free convection flow patterns in the liquid.