Bifurcation Analysis of Cellular Interfaces in Unidirectional Solidification of a Dilute Binary Mixture

Abstract

Analytical bifurcation theory is used to study interface patterns in the unidirectional solidification of a dilute binary mixture. Armbruster and Dangelmayr’s classification of bifurcation equations equivariant under the diagonal action in $\mathbb{C}^2 $ of the symmetry group $O( 2 )$ is used to show that the local bifurcation structure of the model long-wave evolution equation derived by Riley and Davis is equivalent to that of a codimension-one normal form. xamination of the bifurcation structure of the normal form then leads to a description of the development of three-dimensional interface patterns during unidirectional solidification. The general methodology underlying this study is also discussed, and conditions under which the method can be applied to other evolution equations are derived and clearly stated.