A nonlinear stability analysis of a model equation for alloy solidification

Abstract

A weakly nonlinear stability analysis of the Stuart-Watson type is performed on the planar interface solution to the Sivashinsky model equation for dilute binary alloy solidification. The Landau constant appearing in the amplitude equation is calculated by means of both a direct method of solution and the traditional more indirect one of employing the adjoint linear eigenvector. It is concluded that for moving boundary problems of this sort the direct method is superior to the adjoint operator method especially in those instances where the solution itself is desired and not merely just the solvability condition. In addition, the discrepancy cited by Caroli et al. (J. Physique 43 (1982) 1767) between their formula for the Landau constant, calculated by a direct method, and that of Wollkind and Segel (Philos. Trans. R. Soc. London 268 (1970) 351), calculated by an adjoint operator method which should have yeilded identical results, is resolved in favour of the latter authors