A nonlinear stability analysis of the melting 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 melting of a dilute binary alloy. A nonlinear model is used to describe an experimental situation in which melting 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) Solute concentration in both phases is small. (3) The effects of interface attachment kinetics are negligible. (4) The extent of the liquid and solid phases is infinite. (5) D S = D L where D S( D L) is the diffusion coefficient of the solute in the solid (liquid). (6) D / ϰ L ⪡ 1, where D is the common value of the solute diffusion coefficients and ϰ L is the thermal diffusivity in the liquid. (7) ϱ S c S = ϱ L c L where ϱ S( ϱ L) and c S( c L) are the density and specific heat at constant pressure respectively in the solid (liquid). (8) G ≊ G c where G is the absolute value of the imposed temperature gradient in the liquid and G c is the critical value of G at which linear theory predicts the onset of instability. The analysis is expected to be asymptotically valid as G → G c. It is found that the interface can be unstable to finite amplitude disturbances even when linear 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.