Pattern formation in directional solidification under shear flow. I. Linear stability analysis and basic patterns.

An asymptotic interface equation for directional solidification near the absolute stability limit is extended by a nonlocal term describing a shear flow parallel to the interface. In the long-wave limit considered, the flow acts destabilizing on a planar interface. Moreover, linear stability analysis suggests that the morphology diagram is modified by the flow near onset of the Mullins-Sekerka instability. Via numerical analysis, the bifurcation structure of the system is shown to change. Besides the known hexagonal cells, structures consisting of stripes arise. Due to its symmetry-breaking properties, the flow term induces a lateral drift of the whole pattern, once the instability has become active. The drift velocity is measured numerically and described analytically in the framework of a linear analysis. At large flow strength, the linear description breaks down, which is accompanied by a transition to flow-dominated morphologies which is described in the following paper. Small and intermediate flows lead to increased order in the lattice structure of the pattern, facilitating the elimination of defects. Locally oscillating structures appear closer to the instability threshold with flow than without.

Pattern formation in directional solidification under shear flow. II. Morphologies and their characterization.

In the preceding paper, we have established an interface equation for directional solidification under the influence of a shear flow parallel to the interface. This equation is asymptotically valid near the absolute stability limit. The flow, described by a nonlocal term, induces a lateral drift of the whole pattern due to its symmetry-breaking properties. We find that at not-too-large flow strengths, the transcritical nature of the transition to hexagonal patterns shows up via a hexagonal modulation of the stripe pattern even when the linear instability threshold of the flowless case has not yet been attained. When the flow term is large, the linear description of the drift velocity breaks down and transitions to flow-dominated morphologies take place. The competition between flow-induced and diffusion-induced patterns (controlled by the temperature gradient) leads to new phenomena such as the transition to a different lattice structure in an array of hexagonal cells. Several methods to characterize the morphologies and their transitions are investigated and compared. In particular, we consider two different ways of defining topological defects useful in the description of patterns and we discuss how they are related to each other.