A nonlinear stability analysis of the freezing of a dilute binary alloy

Abstract
An investigation is made of the stability of the shape of a moving planar interface between the liquid and solid phases in the freezing of a dilute binary alloy. A nonlinear model is used to describe an experimental situation in which solidification is controlled so that the mean position of the interface moves with constant speed. The model postulates two-dimensional diffusion of solute and heat such that: 1. Convection in the liquid is negligible. 2. Diffusion of the solute in the solid is negligible. 3. Solute concentration in the liquid is small. 4. The effects of interface attachment kinetics are negligible. 5. The extent of the liquid and solid phases is infinite. 6. c8=c1 where c8(c1) is the specific heat per unit volume of the solid (liquid). 7. (D/Dth) 1, , where D is the diffusion coefficient of the solute in the liquid and Dth is the thermal diffusivity in the liquid. 8. G= G where G is the imposed temperature gradient in the liquid and Gcis the critical value of G at which linear theory predicts the onset of instability. The analysis is expected to be asymptotically valid as G-Gc. It is found that the interface can be unstable to finite amplitude disturbances even when linear stability theory predicts stability to infinitesimal disturbances. Further, cellular structure can be anticipated for certain ranges of parameter values. These results are in accord with relevant experimental evidence.

This publication has 9 references indexed in Scilit: