VI-2

by Yi-Ju Chen and Stephen H. Davis

Northwestern University

A steady cellular convection modifies the morphological instability of a binary alloy that undergoes directional solidification. When the convection wavelength is far longer than that of the morphological cells, the behaviour of the moving front is described by a slow, spatial--temporal dynamics obtained through a multiple-scale analysis. The resulting system has a parametric-excitation structure in space, with complex parameters characterizing the interactions between flow, solute diffusion, and rejection. The convection in general stabilizes two-dimensional disturbances orientated with the flow, but destabilizes three-dimensional disturbances. When the flow is weak, the morphological instability behaves incommensurably to the flow wavelength, but as the flow gets stronger, the instability becomes quantized and forced to fit into the flow-box. At large flow strength the instability is localized, confined in narrow envelopes. In this case the solutions are discrete eigenstates in an unbounded space. Their stability boundaries and asymptotics are obtained

by the WKB analysis. The weakly nonlinear interaction is delivered through the Lyapunov--Schmidt method.