Bifurcation analysis of solutal convection during directional solidification

Abstract

Techniques of bifurcation theory are used to analyze the possible forms of steady, two-dimensional, buoyancy-driven flow present during the vertical directional solidification of a dilute binary mixture. Morphological instabilities of the crystal–melt interface are ignored and the interface is assumed to be planar. The focus of the study concerns the effect of lateral confinement on the flow patterns observable in this simple model system. The use of a time-dependent numerical scheme by McFadden and Coriell [Proceedings of the First National Fluid Dynamics Congress, Cincinnati (AIAA, Washington, DC, 1988), p. 1572] to investigate this same problem resulted in gaps in the bifurcation structure, because unstable steady states exist that cannot be computed by their method. The present work extends and completes the description of the possible steady states. In particular, the complex modal exchanges that occur as the separation between the lateral walls varies are elucidated, and stable flow regimes and the sensitivity of the results to the closure condition at the upper boundary are determined.