Direct numerical simulations of layer solidification for unsteady condition have been carried out for a eutectic binary fluid of lauric acid and myristic acid, which exhibits a laminar flow between two parallel walls of different temperatures, Two-dimensional differential equations for momentum, heat and mass transfer are solved simultaneously using non-orthogonal coordinate systems. The boundary-fit grid is generated according to the time-changed shape of the solid-liquid interface. The predictions are discussed with reference to the time-changes of the shape of the solid-liquid (S-L) interface, the growth rate, the concentration at the S-L interface, the super-cooled layer adjacent to the S-L interface, and the concentration in the solid.