A numerical simulation study has been carried out to examine the effect of a static magnetic field on the solidification process of an alloy. A mathematical model, based on the continuum model, was developed for the computation of a transient double-diffusive fluid flow under Lorentz body force. The model includes conservation of mass and momentum, heat, species and electrical charge balance equations. The simulation domain was selected as a cavity filled with a metallic alloy and differentially heated, which may be taken as a Bridgman model domain used in the crystal growth process. The solution is carried out by using a Finite Volume Method. Study of the direction and the intensity of the applied magnetic field effects on stabilizing the double diffusive flow field were also carried out. Simulation results indicate that the use of a static, magnetic field in this growth setup is effective in suppressing natural convection in the solution.