a Nonlinear Analysis and Numerical Simulation of Directional Solidification in Binary Alloys.

A nonlinear model for the directional solidification of binary. alloy systems is studied. Using Green's function methods, an. integro differential equation for the interface motion is obtained. A systematic expansion in powers of the interfacial displacement. is carried out in order to investigate the time evolution of a system. just beyond the neutral stability point. The theory involves five. independent parameters: (nu) proportional to the inverse of the(' ). average temperature gradient, (xi) proportional to the capillary. length, the latent heat generated at the interface (lamda), the thermal. conductivity ratio in the two phases (beta) and the partition coefficient. K. It was found that one mode stationary states existed only for a. small range in parameter space. Also the two mode states were. found to exhibit gaps in Fourier space as the value of (nu) is increased(' ). beyond the critical point. An elaborate numerical simulation involving an implicit scheme and interface tracking methods was also carried out. Excellent agreement was obtained with theoretical amplification rates.